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## G = C4×C3.S4order 288 = 25·32

### Direct product of C4 and C3.S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3.A4 — C4×C3.S4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4 — C2×C3.S4 — C4×C3.S4
 Lower central C3.A4 — C4×C3.S4
 Upper central C1 — C4

Generators and relations for C4×C3.S4
G = < a,b,c,d,e,f | a4=b3=c2=d2=f2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=b-1e2 >

Subgroups: 514 in 96 conjugacy classes, 20 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, D9, C18, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4×D4, Dic9, C36, C3.A4, D18, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, S3×C2×C4, C2×C3⋊D4, C22×C12, C4×D9, C3.S4, C2×C3.A4, C4×C3⋊D4, C6.S4, C4×C3.A4, C2×C3.S4, C4×C3.S4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, D9, C4×S3, S4, D18, C2×S4, C4×D9, C3.S4, C4×S4, C2×C3.S4, C4×C3.S4

Smallest permutation representation of C4×C3.S4
On 36 points
Generators in S36
(1 11 20 34)(2 12 21 35)(3 13 22 36)(4 14 23 28)(5 15 24 29)(6 16 25 30)(7 17 26 31)(8 18 27 32)(9 10 19 33)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)
(1 20)(2 21)(4 23)(5 24)(7 26)(8 27)(11 34)(12 35)(14 28)(15 29)(17 31)(18 32)
(2 21)(3 22)(5 24)(6 25)(8 27)(9 19)(10 33)(12 35)(13 36)(15 29)(16 30)(18 32)
(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 33 34 35 36)
(1 20)(2 19)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 36)

G:=sub<Sym(36)| (1,11,20,34)(2,12,21,35)(3,13,22,36)(4,14,23,28)(5,15,24,29)(6,16,25,30)(7,17,26,31)(8,18,27,32)(9,10,19,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,20)(2,21)(4,23)(5,24)(7,26)(8,27)(11,34)(12,35)(14,28)(15,29)(17,31)(18,32), (2,21)(3,22)(5,24)(6,25)(8,27)(9,19)(10,33)(12,35)(13,36)(15,29)(16,30)(18,32), (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)(28,29,30,31,32,33,34,35,36), (1,20)(2,19)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36)>;

G:=Group( (1,11,20,34)(2,12,21,35)(3,13,22,36)(4,14,23,28)(5,15,24,29)(6,16,25,30)(7,17,26,31)(8,18,27,32)(9,10,19,33), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)(28,31,34)(29,32,35)(30,33,36), (1,20)(2,21)(4,23)(5,24)(7,26)(8,27)(11,34)(12,35)(14,28)(15,29)(17,31)(18,32), (2,21)(3,22)(5,24)(6,25)(8,27)(9,19)(10,33)(12,35)(13,36)(15,29)(16,30)(18,32), (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)(28,29,30,31,32,33,34,35,36), (1,20)(2,19)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,36) );

G=PermutationGroup([[(1,11,20,34),(2,12,21,35),(3,13,22,36),(4,14,23,28),(5,15,24,29),(6,16,25,30),(7,17,26,31),(8,18,27,32),(9,10,19,33)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27),(28,31,34),(29,32,35),(30,33,36)], [(1,20),(2,21),(4,23),(5,24),(7,26),(8,27),(11,34),(12,35),(14,28),(15,29),(17,31),(18,32)], [(2,21),(3,22),(5,24),(6,25),(8,27),(9,19),(10,33),(12,35),(13,36),(15,29),(16,30),(18,32)], [(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),(28,29,30,31,32,33,34,35,36)], [(1,20),(2,19),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,36)]])

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E ··· 4J 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A 18B 18C 36A ··· 36F order 1 2 2 2 2 2 3 4 4 4 4 4 ··· 4 6 6 6 9 9 9 12 12 12 12 18 18 18 36 ··· 36 size 1 1 3 3 18 18 2 1 1 3 3 18 ··· 18 2 6 6 8 8 8 2 2 6 6 8 8 8 8 ··· 8

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 6 6 6 type + + + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D6 D9 C4×S3 D18 C4×D9 S4 C2×S4 C4×S4 C3.S4 C2×C3.S4 C4×C3.S4 kernel C4×C3.S4 C6.S4 C4×C3.A4 C2×C3.S4 C3.S4 C22×C12 C22×C6 C22×C4 C2×C6 C23 C22 C12 C6 C3 C4 C2 C1 # reps 1 1 1 1 4 1 1 3 2 3 6 2 2 4 1 1 2

Matrix representation of C4×C3.S4 in GL5(𝔽37)

 6 0 0 0 0 0 6 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36
,
 36 36 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 1 1 0 0 0 0 0 36
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 36 0 1
,
 26 6 0 0 0 31 20 0 0 0 0 0 1 0 35 0 0 0 0 1 0 0 0 36 36
,
 1 0 0 0 0 36 36 0 0 0 0 0 36 35 0 0 0 0 1 0 0 0 0 36 36

G:=sub<GL(5,GF(37))| [6,0,0,0,0,0,6,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[36,1,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,1,0,0,0,0,1,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,36,0,0,0,36,0,0,0,0,0,1],[26,31,0,0,0,6,20,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,35,1,36],[1,36,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,35,1,36,0,0,0,0,36] >;

C4×C3.S4 in GAP, Magma, Sage, TeX

C_4\times C_3.S_4
% in TeX

G:=Group("C4xC3.S4");
// GroupNames label

G:=SmallGroup(288,333);
// by ID

G=gap.SmallGroup(288,333);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-3,-2,2,36,1123,192,1684,6053,782,3534,1350]);
// Polycyclic

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

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