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G = C22×GL2(𝔽3)  order 192 = 26·3

Direct product of C22 and GL2(𝔽3)

direct product, non-abelian, soluble

Aliases: C22×GL2(𝔽3), C23.21S4, SL2(𝔽3)⋊1C23, (C2×Q8)⋊2D6, Q8⋊(C22×S3), C2.9(C22×S4), (C22×Q8)⋊3S3, C22.26(C2×S4), (C2×SL2(𝔽3))⋊4C22, (C22×SL2(𝔽3))⋊5C2, SmallGroup(192,1475)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C22×GL2(𝔽3)
C1C2Q8SL2(𝔽3)GL2(𝔽3)C2×GL2(𝔽3) — C22×GL2(𝔽3)
SL2(𝔽3) — C22×GL2(𝔽3)
C1C23

Generators and relations for C22×GL2(𝔽3)
 G = < a,b,c,d,e,f | a2=b2=c4=e3=f2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=fdf=c-1, ece-1=cd, fcf=c2d, ede-1=c, fef=e-1 >

Subgroups: 875 in 213 conjugacy classes, 37 normal (8 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C22 [×7], C22 [×16], S3 [×8], C6 [×7], C8 [×4], C2×C4 [×6], D4 [×10], Q8, Q8 [×4], C23, C23 [×10], D6 [×28], C2×C6 [×7], C2×C8 [×6], SD16 [×16], C22×C4, C2×D4 [×9], C2×Q8 [×3], C2×Q8 [×3], C24, SL2(𝔽3), C22×S3 [×14], C22×C6, C22×C8, C2×SD16 [×12], C22×D4, C22×Q8, GL2(𝔽3) [×4], C2×SL2(𝔽3) [×3], S3×C23, C22×SD16, C2×GL2(𝔽3) [×6], C22×SL2(𝔽3), C22×GL2(𝔽3)
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D6 [×3], S4, C22×S3, GL2(𝔽3) [×4], C2×S4 [×3], C2×GL2(𝔽3) [×6], C22×S4, C22×GL2(𝔽3)

Smallest permutation representation of C22×GL2(𝔽3)
On 32 points
Generators in S32
(1 24)(2 21)(3 22)(4 23)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)
(1 16)(2 13)(3 14)(4 15)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 11 3 9)(2 10 4 12)(5 32 7 30)(6 31 8 29)(13 20 15 18)(14 19 16 17)(21 28 23 26)(22 27 24 25)
(2 11 10)(4 9 12)(5 8 31)(6 29 7)(13 17 20)(15 19 18)(21 25 28)(23 27 26)
(1 16)(2 19)(3 14)(4 17)(5 23)(6 28)(7 21)(8 26)(9 13)(10 18)(11 15)(12 20)(22 32)(24 30)(25 29)(27 31)

G:=sub<Sym(32)| (1,24)(2,21)(3,22)(4,23)(5,17)(6,18)(7,19)(8,20)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30), (1,16)(2,13)(3,14)(4,15)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,11,3,9)(2,10,4,12)(5,32,7,30)(6,31,8,29)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25), (2,11,10)(4,9,12)(5,8,31)(6,29,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26), (1,16)(2,19)(3,14)(4,17)(5,23)(6,28)(7,21)(8,26)(9,13)(10,18)(11,15)(12,20)(22,32)(24,30)(25,29)(27,31)>;

G:=Group( (1,24)(2,21)(3,22)(4,23)(5,17)(6,18)(7,19)(8,20)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30), (1,16)(2,13)(3,14)(4,15)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(21,31)(22,32)(23,29)(24,30), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,11,3,9)(2,10,4,12)(5,32,7,30)(6,31,8,29)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25), (2,11,10)(4,9,12)(5,8,31)(6,29,7)(13,17,20)(15,19,18)(21,25,28)(23,27,26), (1,16)(2,19)(3,14)(4,17)(5,23)(6,28)(7,21)(8,26)(9,13)(10,18)(11,15)(12,20)(22,32)(24,30)(25,29)(27,31) );

G=PermutationGroup([(1,24),(2,21),(3,22),(4,23),(5,17),(6,18),(7,19),(8,20),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30)], [(1,16),(2,13),(3,14),(4,15),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,17),(12,18),(21,31),(22,32),(23,29),(24,30)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,11,3,9),(2,10,4,12),(5,32,7,30),(6,31,8,29),(13,20,15,18),(14,19,16,17),(21,28,23,26),(22,27,24,25)], [(2,11,10),(4,9,12),(5,8,31),(6,29,7),(13,17,20),(15,19,18),(21,25,28),(23,27,26)], [(1,16),(2,19),(3,14),(4,17),(5,23),(6,28),(7,21),(8,26),(9,13),(10,18),(11,15),(12,20),(22,32),(24,30),(25,29),(27,31)])

32 conjugacy classes

class 1 2A···2G2H2I2J2K 3 4A4B4C4D6A···6G8A···8H
order12···22222344446···68···8
size11···112121212866668···86···6

32 irreducible representations

dim111222334
type++++++++
imageC1C2C2S3D6GL2(𝔽3)S4C2×S4GL2(𝔽3)
kernelC22×GL2(𝔽3)C2×GL2(𝔽3)C22×SL2(𝔽3)C22×Q8C2×Q8C22C23C22C22
# reps161138264

Matrix representation of C22×GL2(𝔽3) in GL6(𝔽73)

7200000
0720000
0072000
0007200
0000720
0000072
,
100000
010000
0072000
0007200
0000720
0000072
,
100000
010000
001000
000100
00005352
00003320
,
100000
010000
001000
000100
00005340
00002120
,
0720000
1720000
0007200
0017200
0000072
0000172
,
010000
100000
0007200
0072000
0000072
0000720

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,33,0,0,0,0,52,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,53,21,0,0,0,0,40,20],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

C22×GL2(𝔽3) in GAP, Magma, Sage, TeX

C_2^2\times {\rm GL}_2({\mathbb F}_3)
% in TeX

G:=Group("C2^2xGL(2,3)");
// GroupNames label

G:=SmallGroup(192,1475);
// by ID

G=gap.SmallGroup(192,1475);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^4=e^3=f^2=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*d*f=c^-1,e*c*e^-1=c*d,f*c*f=c^2*d,e*d*e^-1=c,f*e*f=e^-1>;
// generators/relations

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