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## G = C2×C4.6S4order 192 = 26·3

### Direct product of C2 and C4.6S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — SL2(𝔽3) — C2×C4.6S4
 Chief series C1 — C2 — Q8 — SL2(𝔽3) — GL2(𝔽3) — C2×GL2(𝔽3) — C2×C4.6S4
 Lower central SL2(𝔽3) — C2×C4.6S4
 Upper central C1 — C2×C4

Generators and relations for C2×C4.6S4
G = < a,b,c,d,e,f | a2=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 539 in 153 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, SL2(𝔽3), C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C2×C4○D4, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), C4.A4, S3×C2×C4, C2×C4○D8, C2×CSU2(𝔽3), C2×GL2(𝔽3), C4.6S4, C2×C4.A4, C2×C4.6S4
Quotients: C1, C2, C22, S3, C23, D6, S4, C22×S3, C2×S4, C4.6S4, C22×S4, C2×C4.6S4

Smallest permutation representation of C2×C4.6S4
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 23)(6 24)(7 21)(8 22)(9 17)(10 18)(11 19)(12 20)(13 31)(14 32)(15 29)(16 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 13 3 15)(2 14 4 16)(5 20 7 18)(6 17 8 19)(9 22 11 24)(10 23 12 21)(25 31 27 29)(26 32 28 30)
(1 17 3 19)(2 18 4 20)(5 14 7 16)(6 15 8 13)(9 27 11 25)(10 28 12 26)(21 30 23 32)(22 31 24 29)
(5 14 18)(6 15 19)(7 16 20)(8 13 17)(9 22 31)(10 23 32)(11 24 29)(12 21 30)
(5 16)(6 13)(7 14)(8 15)(9 11)(10 12)(17 19)(18 20)(21 32)(22 29)(23 30)(24 31)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A ··· 8H 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 8 ··· 8 12 12 12 12 size 1 1 1 1 6 6 12 12 8 1 1 1 1 6 6 12 12 8 8 8 6 ··· 6 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 3 3 3 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 C4.6S4 S4 C2×S4 C2×S4 C4.6S4 kernel C2×C4.6S4 C2×CSU2(𝔽3) C2×GL2(𝔽3) C4.6S4 C2×C4.A4 C2×C4○D4 C2×Q8 C4○D4 C2 C2×C4 C4 C22 C2 # reps 1 1 1 4 1 1 1 2 8 2 4 2 4

Matrix representation of C2×C4.6S4 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 72 0 0 0 0 72
,
 1 0 0 0 0 1 0 0 0 0 27 0 0 0 0 27
,
 1 0 0 0 0 1 0 0 0 0 20 33 0 0 52 53
,
 1 0 0 0 0 1 0 0 0 0 53 52 0 0 33 20
,
 0 72 0 0 1 72 0 0 0 0 20 20 0 0 41 52
,
 0 1 0 0 1 0 0 0 0 0 1 72 0 0 0 72
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,20,52,0,0,33,53],[1,0,0,0,0,1,0,0,0,0,53,33,0,0,52,20],[0,1,0,0,72,72,0,0,0,0,20,41,0,0,20,52],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,72,72] >;

C2×C4.6S4 in GAP, Magma, Sage, TeX

C_2\times C_4._6S_4
% in TeX

G:=Group("C2xC4.6S4");
// GroupNames label

G:=SmallGroup(192,1480);
// by ID

G=gap.SmallGroup(192,1480);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,520,451,1684,655,172,1013,404,285,124]);
// Polycyclic

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

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