Over any field K, linear group usually refers to an algebraic group which is a (Zariski closed) subgroup of the general linear group GL(n,K) of invertible nxn matrices with coefficients in K. Similarly, projective linear group refers to a closed subgroup of PGL(n,K)=GL(n,K)/{scalar matrices}, the group of linear automorphisms of the projective n-space. When K=Fq is a finite field, these groups are finite, and often simple or almost simple, and so they play an important role in the classification of finite simple groups.
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
S3 | GL2(F2) | General linear group on F22 | S3 | 3 | 2+ | 6,1 |
SL2(F2) | Special linear group on F22 | |||||
PGL2(F2) | Projective linear group on F22 | |||||
PSL2(F2) | Projective special linear group on F22 | |||||
SO3(F2) | Special orthogonal group on F23 | |||||
SU2(F2) | Special unitary group on F22 | |||||
O3(F2) | Orthogonal group on F23 | |||||
PSO3(F2) | Projective special orthogonal group on F23 | |||||
PO3(F2) | Projective orthogonal group on F23 | |||||
AGL1(F3) | Affine linear group on F31 | |||||
PU2(F2) | Projective unitary group on F22 | |||||
PSU2(F2) | Projective special unitary group on F22 | |||||
CO3(F2) | Conformal orthogonal group on F23 | |||||
CSU2(F2) | Conformal special unitary group on F22 | |||||
CSO3(F2) | Conformal special orthogonal group on F23 | |||||
Spin3(F2) | Spin group on F23 | |||||
Ω3(F2) | Omega group on F23 | |||||
PΩ3(F2) | Projective Omega group on F23 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
D4 | AΣL1(F4) | Affine special semilinear group on F41 | D4 | 4 | 2+ | 8,3 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
A4 | PSL2(F3) | Projective special linear group on F32 | A4 | 4 | 3+ | 12,3 |
AGL1(F4) | Affine linear group on F41 | |||||
PSU2(F3) | Projective special unitary group on F32 | |||||
Ω3(F3) | Omega group on F33 | |||||
PΩ3(F3) | Projective Omega group on F33 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C3xS3 | U2(F2) | Unitary group on F22 | C3xS3 | 6 | 2 | 18,3 |
AΣL1(F9) | Affine special semilinear group on F91 | |||||
CU2(F2) | Conformal unitary group on F22 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F5 | AGL1(F5) | Affine linear group on F51 | F5 | 5 | 4+ | 20,3 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
SL2(F3) | SL2(F3) | Special linear group on F32 | SL(2,3) | 8 | 2- | 24,3 |
SU2(F3) | Special unitary group on F32 | |||||
Spin3(F3) | Spin group on F33 | |||||
S4 | PGL2(F3) | Projective linear group on F32 | S4 | 4 | 3+ | 24,12 |
SO3(F3) | Special orthogonal group on F33 | |||||
PSO3(F3) | Projective special orthogonal group on F33 | |||||
PO3(F3) | Projective orthogonal group on F33 | |||||
AGL2(F2) | Affine linear group on F22 | |||||
ASL2(F2) | Affine special linear group on F22 | |||||
AΓL1(F4) | Affine semilinear group on F41 | |||||
PU2(F3) | Projective unitary group on F32 | |||||
CSO3(F3) | Conformal special orthogonal group on F33 | |||||
C2xA4 | AΣL1(F8) | Affine special semilinear group on F81 | C2xA4 | 6 | 3+ | 24,13 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
S32 | Spin4+ (F2) | Spin group of + type on F24 | S3^2 | 6 | 4+ | 36,10 |
Ω4+ (F2) | Omega group of + type on F24 | |||||
PΩ4+ (F2) | Projective Omega group of + type on F24 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F7 | AGL1(F7) | Affine linear group on F71 | F7 | 7 | 6+ | 42,1 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
CSU2(F3) | CSU2(F3) | Conformal special unitary group on F32 | CSU(2,3) | 16 | 2- | 48,28 |
GL2(F3) | GL2(F3) | General linear group on F32 | GL(2,3) | 8 | 2 | 48,29 |
C2xS4 | O3(F3) | Orthogonal group on F33 | C2xS4 | 6 | 3+ | 48,48 |
CO3(F3) | Conformal orthogonal group on F33 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C5xD5 | AΣL1(F25) | Affine special semilinear group on F251 | C5xD5 | 10 | 2 | 50,3 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F8 | AGL1(F8) | Affine linear group on F81 | F8 | 8 | 7+ | 56,11 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
A5 | SL2(F4) | Special linear group on F42 | A5 | 5 | 3+ | 60,5 |
PGL2(F4) | Projective linear group on F42 | |||||
PSL2(F5) | Projective special linear group on F52 | |||||
PSL2(F4) | Projective special linear group on F42 | |||||
SO3(F4) | Special orthogonal group on F43 | |||||
SU2(F4) | Special unitary group on F42 | |||||
O3(F4) | Orthogonal group on F43 | |||||
PSO3(F4) | Projective special orthogonal group on F43 | |||||
PO3(F4) | Projective orthogonal group on F43 | |||||
PU2(F4) | Projective unitary group on F42 | |||||
PSU2(F5) | Projective special unitary group on F52 | |||||
PSU2(F4) | Projective special unitary group on F42 | |||||
CSU2(F4) | Conformal special unitary group on F42 | |||||
Spin3(F4) | Spin group on F43 | |||||
Ω3(F5) | Omega group on F53 | |||||
Ω3(F4) | Omega group on F43 | |||||
Ω4- (F2) | Omega group of - type on F24 | |||||
PΩ3(F4) | Projective Omega group on F43 | |||||
PΩ3(F5) | Projective Omega group on F53 | |||||
PΩ4- (F2) | Projective Omega group of - type on F24 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C2wrC4 | AΣL1(F16) | Affine special semilinear group on F161 | C2wrC4 | 8 | 4+ | 64,32 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F9 | AGL1(F9) | Affine linear group on F91 | F9 | 9 | 8+ | 72,39 |
S3wrC2 | SO4+ (F2) | Special orthogonal group of + type on F24 | S3wrC2 | 6 | 4+ | 72,40 |
O4+ (F2) | Orthogonal group of + type on F24 | |||||
PSO4+ (F2) | Projective special orthogonal group of + type on F24 | |||||
PO4+ (F2) | Projective orthogonal group of + type on F24 | |||||
CO4+ (F2) | Conformal orthogonal group of + type on F24 | |||||
CSO4+ (F2) | Conformal special orthogonal group of + type on F24 | |||||
PCO4+ (F2) | Projective conformal orthogonal group of + type on F24 | |||||
PCSO4+ (F2) | Projective conformal special orthogonal group of + type on F24 | |||||
PSU3(F2) | PSU3(F2) | Projective special unitary group on F23 | PSU(3,2) | 9 | 8+ | 72,41 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C3wrC3 | AΣL1(F27) | Affine special semilinear group on F271 | C3wrC3 | 9 | 3 | 81,7 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
U2(F3) | U2(F3) | Unitary group on F32 | U(2,3) | 24 | 2 | 96,67 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C7xD7 | AΣL1(F49) | Affine special semilinear group on F491 | C7xD7 | 14 | 2 | 98,3 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F11 | AGL1(F11) | Affine linear group on F111 | F11 | 11 | 10+ | 110,1 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
AΓL1(F9) | AΓL1(F9) | Affine semilinear group on F91 | AGammaL(1,9) | 9 | 8+ | 144,182 |
A42 | PΩ4+ (F3) | Projective Omega group of + type on F34 | A4^2 | 12 | 9+ | 144,184 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F13 | AGL1(F13) | Affine linear group on F131 | F13 | 13 | 12+ | 156,7 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C2xC24:C5 | AΣL1(F32) | Affine special semilinear group on F321 | C2xC2^4:C5 | 10 | 5+ | 160,235 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
GL3(F2) | GL3(F2) | General linear group on F23 | GL(3,2) | 7 | 3 | 168,42 |
SL3(F2) | Special linear group on F23 | |||||
PGL3(F2) | Projective linear group on F23 | |||||
PSL3(F2) | Projective special linear group on F23 | |||||
PSL2(F7) | Projective special linear group on F72 | |||||
PSU2(F7) | Projective special unitary group on F72 | |||||
Ω3(F7) | Omega group on F73 | |||||
PΩ3(F7) | Projective Omega group on F73 | |||||
AΓL1(F8) | AΓL1(F8) | Affine semilinear group on F81 | AGammaL(1,8) | 8 | 7+ | 168,43 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C3xA5 | GL2(F4) | General linear group on F42 | C3xA5 | 15 | 3 | 180,19 |
CO3(F4) | Conformal orthogonal group on F43 | |||||
CSO3(F4) | Conformal special orthogonal group on F43 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
CU2(F3) | CU2(F3) | Conformal unitary group on F32 | CU(2,3) | 32 | 2 | 192,963 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
SU3(F2) | SU3(F2) | Special unitary group on F23 | SU(3,2) | 27 | 3 | 216,88 |
CSU3(F2) | Conformal special unitary group on F23 | |||||
ASL2(F3) | ASL2(F3) | Affine special linear group on F32 | ASL(2,3) | 9 | 8+ | 216,153 |
PU3(F2) | Projective unitary group on F23 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
CSU2(F5) | CSU2(F5) | Conformal special unitary group on F52 | CSU(2,5) | 48 | 4- | 240,89 |
C2xS5 | O3(F5) | Orthogonal group on F53 | C2xS5 | 10 | 4+ | 240,189 |
F16 | AGL1(F16) | Affine linear group on F161 | F16 | 16 | 15+ | 240,191 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C11xD11 | AΣL1(F121) | Affine special semilinear group on F1211 | C11xD11 | 22 | 2 | 242,3 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F17 | AGL1(F17) | Affine linear group on F171 | F17 | 17 | 16+ | 272,50 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
Ω4+ (F3) | Ω4+ (F3) | Omega group of + type on F34 | Omega+(4,3) | 24 | 4+ | 288,860 |
PSO4+ (F3) | PSO4+ (F3) | Projective special orthogonal group of + type on F34 | PSO+(4,3) | 12 | 9+ | 288,1026 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C5xA5 | U2(F4) | Unitary group on F42 | C5xA5 | 25 | 3 | 300,22 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C3xC33:C4 | AΣL1(F81) | Affine special semilinear group on F811 | C3xC3^3:C4 | 12 | 4 | 324,162 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
SL2(F7) | SL2(F7) | Special linear group on F72 | SL(2,7) | 16 | 4 | 336,114 |
SU2(F7) | Special unitary group on F72 | |||||
Spin3(F7) | Spin group on F73 | |||||
PGL2(F7) | PGL2(F7) | Projective linear group on F72 | PGL(2,7) | 8 | 6+ | 336,208 |
SO3(F7) | Special orthogonal group on F73 | |||||
PSO3(F7) | Projective special orthogonal group on F73 | |||||
PO3(F7) | Projective orthogonal group on F73 | |||||
PU2(F7) | Projective unitary group on F72 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C13xD13 | AΣL1(F169) | Affine special semilinear group on F1691 | C13xD13 | 26 | 2 | 338,3 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
F19 | AGL1(F19) | Affine linear group on F191 | F19 | 19 | 18+ | 342,7 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
A6 | PSL2(F9) | Projective special linear group on F92 | A6 | 6 | 5+ | 360,118 |
PSO4- (F3) | Projective special orthogonal group of - type on F34 | |||||
PSU2(F9) | Projective special unitary group on F92 | |||||
Ω3(F9) | Omega group on F93 | |||||
Ω4- (F3) | Omega group of - type on F34 | |||||
PΩ3(F9) | Projective Omega group on F93 | |||||
PΩ4- (F3) | Projective Omega group of - type on F34 | |||||
ΓL2(F4) | ΓL2(F4) | Semilinear group on F42 | GammaL(2,4) | 15 | 6 | 360,120 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
C5xC52:C3 | AΣL1(F125) | Affine special semilinear group on F1251 | C5xC5^2:C3 | 15 | 3 | 375,6 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
AGL2(F3) | AGL2(F3) | Affine linear group on F32 | AGL(2,3) | 9 | 8+ | 432,734 |
Label | d | ρ | ID | |||
---|---|---|---|---|---|---|
GL2(F5) | GL2(F5) | General linear group on F52 | GL(2,5) | 24 | 4 | 480,218 |
C4xS5 | CO3(F5) | Conformal orthogonal group on F53 | C4xS5 | 20 | 4 | 480,943 |