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## G = C6×C3.S4order 432 = 24·33

### Direct product of C6 and C3.S4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C6×C3.S4
G = < a,b,c,d,e,f | a6=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: 622 in 124 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C3×S3, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×C9, C3.A4, C3.A4, D18, C3×Dic3, S3×C6, C62, C62, C2×C3⋊D4, C6×D4, C3×D9, C3×C18, C3.S4, C2×C3.A4, C2×C3.A4, C6×Dic3, C3×C3⋊D4, S3×C2×C6, C2×C62, C3×C3.A4, C6×D9, C2×C3.S4, C6×C3⋊D4, C3×C3.S4, C6×C3.A4, C6×C3.S4
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, D9, C3×S3, S4, D18, S3×C6, C2×S4, C3×D9, C3.S4, C3×S4, C6×D9, C2×C3.S4, C6×S4, C3×C3.S4, C6×C3.S4

Smallest permutation representation of C6×C3.S4
On 36 points
Generators in S36
(1 23 4 26 7 20)(2 24 5 27 8 21)(3 25 6 19 9 22)(10 28 16 34 13 31)(11 29 17 35 14 32)(12 30 18 36 15 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 26)(3 19)(4 20)(6 22)(7 23)(9 25)(10 34)(12 36)(13 28)(15 30)(16 31)(18 33)
(1 26)(2 27)(4 20)(5 21)(7 23)(8 24)(10 34)(11 35)(13 28)(14 29)(16 31)(17 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 11)(2 10)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 36)(26 35)(27 34)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O 6P 6Q 6R 6S 9A ··· 9I 12A 12B 12C 12D 18A ··· 18I order 1 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 9 ··· 9 12 12 12 12 18 ··· 18 size 1 1 3 3 18 18 1 1 2 2 2 18 18 1 1 2 2 2 3 3 3 3 6 ··· 6 18 18 18 18 8 ··· 8 18 18 18 18 8 ··· 8

54 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 6 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 D9 C3×S3 D18 S3×C6 C3×D9 C6×D9 S4 C2×S4 C3×S4 C6×S4 C3.S4 C2×C3.S4 C3×C3.S4 C6×C3.S4 kernel C6×C3.S4 C3×C3.S4 C6×C3.A4 C2×C3.S4 C3.S4 C2×C3.A4 C2×C62 C62 C22×C6 C22×C6 C2×C6 C2×C6 C23 C22 C3×C6 C32 C6 C3 C6 C3 C2 C1 # reps 1 2 1 2 4 2 1 1 3 2 3 2 6 6 2 2 4 4 1 1 2 2

Matrix representation of C6×C3.S4 in GL7(𝔽37)

 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 26 0 0 0 0 0 0 33 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 36
,
 16 0 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0 1 0 0 0 0 0 36 36 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 30 9 0 0 0 0 0 7 7 0 0 0 0 0 0 0 26 20 0 0 0 0 0 31 11 0 0 0 0 0 0 0 0 0 36 0 0 0 0 0 36 0 0 0 0 0 36 0 0

G:=sub<GL(7,GF(37))| [36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[26,33,0,0,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,36],[16,7,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,36,0,0,0,0,0,1,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0],[30,7,0,0,0,0,0,9,7,0,0,0,0,0,0,0,26,31,0,0,0,0,0,20,11,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36,0,0] >;

C6×C3.S4 in GAP, Magma, Sage, TeX

C_6\times C_3.S_4
% in TeX

G:=Group("C6xC3.S4");
// GroupNames label

G:=SmallGroup(432,534);
// by ID

G=gap.SmallGroup(432,534);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,1683,192,2524,9077,782,5298,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=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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