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## G = C3×C6.S4order 432 = 24·33

### Direct product of C3 and C6.S4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C3×C6.S4
G = < a,b,c,d,e,f | a3=b6=c2=d2=1, e3=b2, f2=b3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, ece-1=fcf-1=cd=dc, ede-1=c, df=fd, fef-1=b4e2 >

Subgroups: 314 in 76 conjugacy classes, 22 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C9, C32, Dic3, C12, C2×C6, C2×C6, C22⋊C4, C18, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, C3×C9, Dic9, C3.A4, C3.A4, C3×Dic3, C62, C62, C6.D4, C3×C22⋊C4, C3×C18, C2×C3.A4, C2×C3.A4, C6×Dic3, C2×C62, C3×Dic9, C3×C3.A4, C6.S4, C3×C6.D4, C6×C3.A4, C3×C6.S4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, D9, C3×S3, S4, Dic9, C3×Dic3, A4⋊C4, C3×D9, C3.S4, C3×S4, C3×Dic9, C6.S4, C3×A4⋊C4, C3×C3.S4, C3×C6.S4

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

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J ··· 6O 9A ··· 9I 12A ··· 12H 18A ··· 18I order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 6 6 6 6 6 6 6 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 size 1 1 3 3 1 1 2 2 2 18 18 18 18 1 1 2 2 2 3 3 3 3 6 ··· 6 8 ··· 8 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 C3 C4 C6 C12 S3 Dic3 D9 C3×S3 Dic9 C3×Dic3 C3×D9 C3×Dic9 S4 A4⋊C4 C3×S4 C3×A4⋊C4 C3.S4 C6.S4 C3×C3.S4 C3×C6.S4 kernel C3×C6.S4 C6×C3.A4 C6.S4 C3×C3.A4 C2×C3.A4 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 1 2 2 2 4 1 1 3 2 3 2 6 6 2 2 4 4 1 1 2 2

Matrix representation of C3×C6.S4 in GL5(𝔽37)

 1 0 0 0 0 0 1 0 0 0 0 0 26 0 0 0 0 0 26 0 0 0 0 0 26
,
 11 0 0 0 0 0 27 0 0 0 0 0 36 0 0 0 0 0 36 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 17 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 36 0 0 0 0 21 1 0 0 0 0 0 36
,
 33 0 0 0 0 0 9 0 0 0 0 0 16 35 0 0 0 7 21 1 0 0 30 17 0
,
 0 6 0 0 0 6 0 0 0 0 0 0 9 0 25 0 0 24 6 15 0 0 13 0 28

G:=sub<GL(5,GF(37))| [1,0,0,0,0,0,1,0,0,0,0,0,26,0,0,0,0,0,26,0,0,0,0,0,26],[11,0,0,0,0,0,27,0,0,0,0,0,36,0,0,0,0,0,36,0,0,0,0,0,36],[1,0,0,0,0,0,1,0,0,0,0,0,36,0,17,0,0,0,36,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,36,21,0,0,0,0,1,0,0,0,0,0,36],[33,0,0,0,0,0,9,0,0,0,0,0,16,7,30,0,0,35,21,17,0,0,0,1,0],[0,6,0,0,0,6,0,0,0,0,0,0,9,24,13,0,0,0,6,0,0,0,25,15,28] >;

C3×C6.S4 in GAP, Magma, Sage, TeX

C_3\times C_6.S_4
% in TeX

G:=Group("C3xC6.S4");
// GroupNames label

G:=SmallGroup(432,250);
// by ID

G=gap.SmallGroup(432,250);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,1683,192,2524,9077,782,5298,1350]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^6=c^2=d^2=1,e^3=b^2,f^2=b^3,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^-1=b^-1,e*c*e^-1=f*c*f^-1=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f^-1=b^4*e^2>;
// generators/relations

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