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## G = C6×C32⋊C6order 324 = 22·34

### Direct product of C6 and C32⋊C6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C6×C32⋊C6
 Chief series C1 — C3 — C32 — C33 — C3×He3 — C3×C32⋊C6 — C6×C32⋊C6
 Lower central C32 — C6×C32⋊C6
 Upper central C1 — C6

Generators and relations for C6×C32⋊C6
G = < a,b,c,d | a6=b3=c3=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c-1, dcd-1=c-1 >

Subgroups: 496 in 136 conjugacy classes, 46 normal (19 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C32, C32, C32, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, He3, He3, C33, C33, S3×C6, C2×C3⋊S3, C62, C32⋊C6, C2×He3, C2×He3, S3×C32, C3×C3⋊S3, C32×C6, C32×C6, C3×He3, C2×C32⋊C6, S3×C3×C6, C6×C3⋊S3, C3×C32⋊C6, C6×He3, C6×C32⋊C6
Quotients: C1, C2, C3, C22, S3, C6, C32, D6, C2×C6, C3×S3, C3×C6, S3×C6, C62, C32⋊C6, S3×C32, C2×C32⋊C6, S3×C3×C6, C3×C32⋊C6, C6×C32⋊C6

Smallest permutation representation of C6×C32⋊C6
On 36 points
Generators in S36
(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 19 13)(2 20 14)(3 21 15)(4 22 16)(5 23 17)(6 24 18)(7 31 30)(8 32 25)(9 33 26)(10 34 27)(11 35 28)(12 36 29)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 21 23)(20 22 24)(25 29 27)(26 30 28)(31 35 33)(32 36 34)
(1 34 5 32 3 36)(2 35 6 33 4 31)(7 24)(8 19)(9 20)(10 21)(11 22)(12 23)(13 29 15 25 17 27)(14 30 16 26 18 28)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3T 6A 6B 6C 6D 6E 6F ··· 6K 6L ··· 6T 6U ··· 6AJ order 1 2 2 2 3 3 3 3 3 3 ··· 3 3 ··· 3 6 6 6 6 6 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 9 9 1 1 2 2 2 3 ··· 3 6 ··· 6 1 1 2 2 2 3 ··· 3 6 ··· 6 9 ··· 9

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 6 6 6 6 type + + + + + + + image C1 C2 C2 C3 C3 C6 C6 C6 C6 S3 D6 C3×S3 S3×C6 C32⋊C6 C2×C32⋊C6 C3×C32⋊C6 C6×C32⋊C6 kernel C6×C32⋊C6 C3×C32⋊C6 C6×He3 C2×C32⋊C6 C6×C3⋊S3 C32⋊C6 C2×He3 C3×C3⋊S3 C32×C6 C32×C6 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 2 1 6 2 12 6 4 2 1 1 8 8 1 1 2 2

Matrix representation of C6×C32⋊C6 in GL8(𝔽7)

 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 0 0 6 6 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4
,
 5 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,GF(7))| [3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,6,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[5,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0] >;

C6×C32⋊C6 in GAP, Magma, Sage, TeX

C_6\times C_3^2\rtimes C_6
% in TeX

G:=Group("C6xC3^2:C6");
// GroupNames label

G:=SmallGroup(324,138);
// by ID

G=gap.SmallGroup(324,138);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,2164,1096,7781]);
// Polycyclic

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

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