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## G = C22×GL2(𝔽3)  order 192 = 26·3

### Direct product of C22 and GL2(𝔽3)

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

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C22×GL2(𝔽3)
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3) — C22×GL2(𝔽3)
 Lower central SL2(𝔽3) — C22×GL2(𝔽3)
 Upper central C1 — C23

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 ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 6A ··· 6G 8A ··· 8H order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 6 ··· 6 8 ··· 8 size 1 1 ··· 1 12 12 12 12 8 6 6 6 6 8 ··· 8 6 ··· 6

32 irreducible representations

 dim 1 1 1 2 2 2 3 3 4 type + + + + + + + + image C1 C2 C2 S3 D6 GL2(𝔽3) S4 C2×S4 GL2(𝔽3) kernel C22×GL2(𝔽3) C2×GL2(𝔽3) C22×SL2(𝔽3) C22×Q8 C2×Q8 C22 C23 C22 C22 # reps 1 6 1 1 3 8 2 6 4

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

 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 52 0 0 0 0 33 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 40 0 0 0 0 21 20
,
 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 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

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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