S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) Special unitary group
Group Lie theory
dimℝ 3 executable 7 linked workbenchs
S O ( 3 ) \mathrm{SO}(3) SO ( 3 ) Rotation group in three dimensions
Group Lie theory
dimℝ 3 executable 2 linked workbenchs
S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) Special unitary group
Group Lie theory
dimℝ 8 executable 8 linked workbenchs
P S U ( 3 ) \mathrm{PSU}(3) PSU ( 3 ) Projective special unitary group
Group Lie theory
dimℝ 8 executable 1 linked workbench
U ( 1 ) \mathrm{U}(1) U ( 1 ) Circle group
Group Lie theory
dimℝ 1 reference Sourced record
S p i n ( 4 ) \mathrm{Spin}(4) Spin ( 4 ) Euclidean spin group in four dimensions
Group Lie theory
dimℝ 6 reference Sourced record
S L ( 2 , C ) \mathrm{SL}(2,\mathbb C) SL ( 2 , C ) Lorentz spin group
Group Lie theory
dimℝ 6 reference Sourced record
C 3 \mathrm{C}_{3} C 3 Cyclic group of order three
Group Finite
order 3 executable 1 linked workbench
S 3 \mathrm{S}_{3} S 3 Symmetric group on three letters
Group Finite
order 6 executable 1 linked workbench
A 4 \mathrm{A}_{4} A 4 Alternating group on four letters
Group Finite
order 12 executable 2 linked workbenchs
S 4 \mathrm{S}_{4} S 4 Symmetric group on four letters
Group Finite
order 24 executable 1 linked workbench
A 5 \mathrm{A}_{5} A 5 Alternating group on five letters
Group Finite
order 60 executable 1 linked workbench
Q 8 \mathrm{Q}_{8} Q 8 Quaternion group
Group Finite
order 8 executable 1 linked workbench
D 8 \mathrm{D}_{8} D 8 Dihedral group of order eight
Group Finite
order 8 executable 1 linked workbench
C 2 C_{2} C 2 Cyclic group of order 2
Group Finite
order 2 reference Sourced record
C 4 C_{4} C 4 Cyclic group of order 4
Group Finite
order 4 reference Sourced record
C 5 C_{5} C 5 Cyclic group of order 5
Group Finite
order 5 reference Sourced record
C 6 C_{6} C 6 Cyclic group of order 6
Group Finite
order 6 reference Sourced record
V 4 \mathrm{V}_{4} V 4 Klein four-group
Group Finite
order 4 reference Sourced record
D 10 \mathrm{D}_{10} D 10 Dihedral group of order ten
Group Finite
order 10 reference Sourced record
D 12 \mathrm{D}_{12} D 12 Dihedral group of order twelve
Group Finite
order 12 reference Sourced record
D i c 3 \mathrm{Dic}_{3} Dic 3 Dicyclic group of order twelve
Group Finite
order 12 reference Sourced record
Q 16 \mathrm{Q}_{16} Q 16 Generalized quaternion group of order sixteen
Group Finite
order 16 reference Sourced record
2 T 2T 2 T Binary tetrahedral group
Group Finite
order 24 reference Sourced record
2 O 2O 2 O Binary octahedral group
Group Finite
order 48 reference Sourced record
2 I 2I 2 I Binary icosahedral group
Group Finite
order 120 reference Sourced record
P S L ( 2 , 7 ) \mathrm{PSL}(2,7) PSL ( 2 , 7 ) Projective special linear group PSL(2,7)
Group Finite
order 168 reference Sourced record
Δ ( 27 ) \Delta(27) Δ ( 27 ) Delta(27), the finite Heisenberg group
Group Finite
order 27 reference Sourced record
A 6 \mathrm{A}_{6} A 6 Alternating group on six letters
Group Finite
order 360 reference Sourced record
M 11 \mathrm{M}_{11} M 11 Mathieu group M11
Group Finite
order 7920 classification Sourced record
M 12 \mathrm{M}_{12} M 12 Mathieu group M12
Group Finite
order 95040 classification Sourced record
M 22 \mathrm{M}_{22} M 22 Mathieu group M22
Group Finite
order 443520 classification Sourced record
M 23 \mathrm{M}_{23} M 23 Mathieu group M23
Group Finite
order 10200960 classification Sourced record
M 24 \mathrm{M}_{24} M 24 Mathieu group M24
Group Finite
order 244823040 classification Sourced record
J 1 \mathrm{J}_{1} J 1 First Janko group
Group Finite
order 175560 classification Sourced record
J 2 \mathrm{J}_{2} J 2 Hall–Janko group
Group Finite
order 604800 classification Sourced record
J 3 \mathrm{J}_{3} J 3 Third Janko group
Group Finite
order 50232960 classification Sourced record
J 4 \mathrm{J}_{4} J 4 Fourth Janko group
Group Finite
order 86775571046077562880 classification Sourced record
C o 1 \mathrm{Co}_{1} Co 1 First Conway group
Group Finite
order 4157776806543360000 classification Sourced record
C o 2 \mathrm{Co}_{2} Co 2 Second Conway group
Group Finite
order 42305421312000 classification Sourced record
C o 3 \mathrm{Co}_{3} Co 3 Third Conway group
Group Finite
order 495766656000 classification Sourced record
F i 22 \mathrm{Fi}_{22} Fi 22 Fischer group Fi22
Group Finite
order 64561751654400 classification Sourced record
F i 23 \mathrm{Fi}_{23} Fi 23 Fischer group Fi23
Group Finite
order 4089470473293004800 classification Sourced record
F i 24 ′ \mathrm{Fi}_{24}' Fi 24 ′ Fischer simple group Fi24 prime
Group Finite
order 1255205709190661721292800 classification Sourced record
H S \mathrm{HS} HS Higman–Sims group
Group Finite
order 44352000 classification Sourced record
M c L \mathrm{McL} McL McLaughlin group
Group Finite
order 898128000 classification Sourced record
H e \mathrm{He} He Held group
Group Finite
order 4030387200 classification Sourced record
R u \mathrm{Ru} Ru Rudvalis group
Group Finite
order 145926144000 classification Sourced record
S u z \mathrm{Suz} Suz Sporadic Suzuki group
Group Finite
order 448345497600 classification Sourced record
O ′ N \mathrm{O'N} O ′ N O’Nan group
Group Finite
order 460815505920 classification Sourced record
H N \mathrm{HN} HN Harada–Norton group
Group Finite
order 273030912000000 classification Sourced record
L y \mathrm{Ly} Ly Lyons group
Group Finite
order 51765179004000000 classification Sourced record
T h \mathrm{Th} Th Thompson group
Group Finite
order 90745943887872000 classification Sourced record
B \mathrm{B} B Baby Monster group
Group Finite
order 4154781481226426191177580544000000 classification Sourced record
M \mathrm{M} M Monster group
Group Finite
order 808017424794512875886459904961710757005754368000000000 classification Sourced record
2 F 4 ( 2 ) ′ {}^{2}\!F_{4}(2)' 2 F 4 ( 2 ) ′ Tits group
Group Finite
order 17971200 classification Sourced record
S U ( 4 ) \mathrm{SU}(4) SU ( 4 ) Special unitary group in four dimensions
Group Lie theory
dimℝ 15 executable 4 linked workbenchs
P S U ( 4 ) \mathrm{PSU}(4) PSU ( 4 ) Adjoint compact group of type A₃
Group Lie theory
dimℝ 15 reference Sourced record
S U ( 5 ) \mathrm{SU}(5) SU ( 5 ) Special unitary group in five dimensions
Group Lie theory
dimℝ 24 executable 4 linked workbenchs
U ( 2 ) \mathrm{U}(2) U ( 2 ) Unitary group in two dimensions
Group Lie theory
dimℝ 4 reference Sourced record
S O ( 4 ) \mathrm{SO}(4) SO ( 4 ) Rotation group in four dimensions
Group Lie theory
dimℝ 6 reference Sourced record
S O ( 5 ) \mathrm{SO}(5) SO ( 5 ) Rotation group in five dimensions
Group Lie theory
dimℝ 10 reference Sourced record
S O ( 6 ) \mathrm{SO}(6) SO ( 6 ) Rotation group in six dimensions
Group Lie theory
dimℝ 15 reference Sourced record
S p ( 2 ) \mathrm{Sp}(2) Sp ( 2 ) Compact symplectic group of quaternionic rank two
Group Lie theory
dimℝ 10 executable 1 linked workbench
S p ( 3 ) \mathrm{Sp}(3) Sp ( 3 ) Compact symplectic group of quaternionic rank three
Group Lie theory
dimℝ 21 reference Sourced record
S p i n ( 8 ) \mathrm{Spin}(8) Spin ( 8 ) Simply connected compact spin group in eight dimensions
Group Lie theory
dimℝ 28 executable 1 linked workbench
S O ( 8 ) \mathrm{SO}(8) SO ( 8 ) Rotation group in eight dimensions
Group Lie theory
dimℝ 28 reference Sourced record
P S O ( 8 ) \mathrm{PSO}(8) PSO ( 8 ) Adjoint compact group of type D₄
Group Lie theory
dimℝ 28 reference Sourced record
S p i n ( 10 ) \mathrm{Spin}(10) Spin ( 10 ) Simply connected compact spin group in ten dimensions
Group Lie theory
dimℝ 45 executable 1 linked workbench
S O ( 10 ) \mathrm{SO}(10) SO ( 10 ) Rotation group in ten dimensions
Group Lie theory
dimℝ 45 reference Sourced record
G 2 compact \mathrm{G}_{2}\;\text{compact} G 2 compact Compact group of octonion automorphisms
Group Lie theory
dimℝ 14 executable 1 linked workbench
F 4 compact \mathrm{F}_{4}\;\text{compact} F 4 compact Compact exceptional group of type F₄
Group Lie theory
dimℝ 52 executable 1 linked workbench
E 6 compact, simply connected \mathrm{E}_{6}\;\text{compact, simply connected} E 6 compact, simply connected Simply connected compact exceptional group of type E₆
Group Lie theory
dimℝ 78 executable 1 linked workbench
E 7 compact, simply connected \mathrm{E}_{7}\;\text{compact, simply connected} E 7 compact, simply connected Simply connected compact exceptional group of type E₇
Group Lie theory
dimℝ 133 executable 1 linked workbench
E 8 compact \mathrm{E}_{8}\;\text{compact} E 8 compact Compact exceptional group of type E₈
Group Lie theory
dimℝ 248 executable 1 linked workbench
E 6 compact, adjoint \mathrm{E}_{6}\;\text{compact, adjoint} E 6 compact, adjoint Adjoint compact exceptional group of type E₆
Group Lie theory
dimℝ 78 reference Sourced record
E 7 compact, adjoint \mathrm{E}_{7}\;\text{compact, adjoint} E 7 compact, adjoint Adjoint compact exceptional group of type E₇
Group Lie theory
dimℝ 133 reference Sourced record
S L ( 2 , R ) \mathrm{SL}(2, \mathbb R) SL ( 2 , R ) Real special linear group in two dimensions
Group Lie theory
dimℝ 3 reference Sourced record
P S L ( 2 , R ) \mathrm{PSL}(2, \mathbb R) PSL ( 2 , R ) Adjoint real group of type A₁
Group Lie theory
dimℝ 3 reference Sourced record
S L ( 2 , R ) ~ \widetilde{\mathrm{SL}(2,\mathbb R)} SL ( 2 , R ) Simply connected group integrating sl(2,R)
Group Lie theory
dimℝ 3 reference Sourced record
S L ( 3 , R ) \mathrm{SL}(3, \mathbb R) SL ( 3 , R ) Real special linear group in three dimensions
Group Lie theory
dimℝ 8 reference Sourced record
S U ( 2 , 2 ) \mathrm{SU}(2,2) SU ( 2 , 2 ) Indefinite special unitary group of signature (2,2)
Group Lie theory
dimℝ 15 reference Sourced record
S p ( 4 , R ) \mathrm{Sp}(4, \mathbb R) Sp ( 4 , R ) Real symplectic group on a four-dimensional space
Group Lie theory
dimℝ 10 reference Sourced record
S O 0 ( 1 , 3 ) \mathrm{SO}_0(1,3) SO 0 ( 1 , 3 ) Proper orthochronous Lorentz group in four dimensions
Group Lie theory
dimℝ 6 reference Sourced record
S O 0 ( 2 , 4 ) \mathrm{SO}_0(2,4) SO 0 ( 2 , 4 ) Identity component of the orthogonal group of signature (2,4)
Group Lie theory
dimℝ 15 reference Sourced record
S O 0 ( 2 , 3 ) \mathrm{SO}_0(2,3) SO 0 ( 2 , 3 ) Identity component of the orthogonal group of signature (2,3)
Group Lie theory
dimℝ 10 reference Sourced record
S U ( 3 ) × S U ( 2 ) × U ( 1 ) SU(3)\times SU(2)\times U(1) S U ( 3 ) × S U ( 2 ) × U ( 1 ) Standard Model compact covering product
Group Lie theory
dimℝ 12 executable 3 linked workbenchs
G S M ( 2 ) G_{\mathrm{SM}}^{(2)} G SM ( 2 ) Standard Model global group with diagonal Z2 quotient
Group Lie theory
dimℝ 12 executable 3 linked workbenchs
G S M ( 3 ) G_{\mathrm{SM}}^{(3)} G SM ( 3 ) Standard Model global group with diagonal Z3 quotient
Group Lie theory
dimℝ 12 executable 3 linked workbenchs
G S M ( 6 ) G_{\mathrm{SM}}^{(6)} G SM ( 6 ) Standard Model global group with diagonal Z6 quotient
Group Lie theory
dimℝ 12 executable 3 linked workbenchs