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## G = C3×C6.7S4order 432 = 24·33

### Direct product of C3 and C6.7S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×A4 — C3×C6.7S4
 Chief series C1 — C22 — C2×C6 — C3×A4 — C6×A4 — A4×C3×C6 — C3×C6.7S4
 Lower central C3×A4 — C3×C6.7S4
 Upper central C1 — C6

Generators and relations for C3×C6.7S4
G = < a,b,c,d,e,f | a3=b6=c2=d2=e3=1, 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=e-1 >

Subgroups: 584 in 136 conjugacy classes, 34 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, A4, A4, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×A4, C2×A4, C22×C6, C22×C6, C33, C3×Dic3, C3⋊Dic3, C3×A4, C3×A4, C3×A4, C62, C62, C6.D4, C3×C22⋊C4, A4⋊C4, C32×C6, C6×Dic3, C6×A4, C6×A4, C6×A4, C2×C62, C3×C3⋊Dic3, C32×A4, C3×C6.D4, C3×A4⋊C4, C6.7S4, A4×C3×C6, C3×C6.7S4
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3×S3, C3⋊S3, S4, C3×Dic3, C3⋊Dic3, A4⋊C4, C3×C3⋊S3, C3×S4, C3⋊S4, C3×C3⋊Dic3, C3×A4⋊C4, C6.7S4, C3×C3⋊S4, C3×C6.7S4

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

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

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

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

54 conjugacy classes

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

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 S3 Dic3 Dic3 C3×S3 C3×S3 C3×Dic3 C3×Dic3 S4 A4⋊C4 C3×S4 C3×A4⋊C4 C3⋊S4 C6.7S4 C3×C3⋊S4 C3×C6.7S4 kernel C3×C6.7S4 A4×C3×C6 C6.7S4 C32×A4 C6×A4 C3×A4 C6×A4 C2×C62 C3×A4 C62 C2×A4 C22×C6 A4 C2×C6 C3×C6 C32 C6 C3 C6 C3 C2 C1 # reps 1 1 2 2 2 4 3 1 3 1 6 2 6 2 2 2 4 4 1 1 2 2

Matrix representation of C3×C6.7S4 in GL5(𝔽13)

 1 0 0 0 0 0 1 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 4 0 0 0 0 0 10 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 12 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 12 12
,
 3 0 0 0 0 0 9 0 0 0 0 0 12 12 11 0 0 1 0 0 0 0 0 0 1
,
 0 8 0 0 0 8 0 0 0 0 0 0 5 5 10 0 0 0 8 0 0 0 0 0 8

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[4,0,0,0,0,0,10,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,12,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,12,0,0,0,0,12],[3,0,0,0,0,0,9,0,0,0,0,0,12,1,0,0,0,12,0,0,0,0,11,0,1],[0,8,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,5,8,0,0,0,10,0,8] >;

C3×C6.7S4 in GAP, Magma, Sage, TeX

C_3\times C_6._7S_4
% in TeX

G:=Group("C3xC6.7S4");
// GroupNames label

G:=SmallGroup(432,618);
// by ID

G=gap.SmallGroup(432,618);
# by ID

G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,675,2524,9077,782,5298,1350]);
// Polycyclic

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

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