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## G = C3×C32⋊C12order 324 = 22·34

### Direct product of C3 and C32⋊C12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C32⋊C12
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×He3 — C3×C32⋊C12
 Lower central C32 — C3×C32⋊C12
 Upper central C1 — C6

Generators and relations for C3×C32⋊C12
G = < a,b,c,d | a3=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=c-1 >

Subgroups: 304 in 96 conjugacy classes, 34 normal (19 characteristic)
C1, C2, C3, C3, C4, C6, C6, C32, C32, C32, Dic3, C12, C3×C6, C3×C6, C3×C6, He3, He3, C33, C33, C3×Dic3, C3⋊Dic3, C3×C12, C2×He3, C2×He3, C32×C6, C32×C6, C3×He3, C32⋊C12, C32×Dic3, C3×C3⋊Dic3, C6×He3, C3×C32⋊C12
Quotients: C1, C2, C3, C4, S3, C6, C32, Dic3, C12, C3×S3, C3×C6, C3×Dic3, C3×C12, C32⋊C6, S3×C32, C32⋊C12, C32×Dic3, C3×C32⋊C6, C3×C32⋊C12

Smallest permutation representation of C3×C32⋊C12
On 36 points
Generators in S36
(1 8 9)(2 5 10)(3 6 11)(4 7 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)
(1 24 28)(2 25 21)(3 18 34)(4 31 15)(5 29 17)(6 14 26)(7 35 23)(8 20 32)(9 16 36)(10 33 13)(11 22 30)(12 27 19)
(1 9 8)(2 5 10)(3 11 6)(4 7 12)(13 21 17)(14 18 22)(15 23 19)(16 20 24)(25 29 33)(26 34 30)(27 31 35)(28 36 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)

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

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

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

60 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 3F ··· 3K 3L ··· 3T 4A 4B 6A 6B 6C 6D 6E 6F ··· 6K 6L ··· 6T 12A ··· 12P order 1 2 3 3 3 3 3 3 ··· 3 3 ··· 3 4 4 6 6 6 6 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 3 ··· 3 6 ··· 6 9 9 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 C3 C3 C4 C6 C6 C12 C12 S3 Dic3 C3×S3 C3×Dic3 C32⋊C6 C32⋊C12 C3×C32⋊C6 C3×C32⋊C12 kernel C3×C32⋊C12 C6×He3 C32⋊C12 C3×C3⋊Dic3 C3×He3 C2×He3 C32×C6 He3 C33 C32×C6 C33 C3×C6 C32 C6 C3 C2 C1 # reps 1 1 6 2 2 6 2 12 4 1 1 8 8 1 1 2 2

Matrix representation of C3×C32⋊C12 in GL8(𝔽13)

 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 3
,
 12 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 9 0 3 0 3 0 0 0 0 9 0 3 0 0 0 9 0 0 10 10 10 0 0 0 0 0 3 0 0 0 0 0 0 0 6 10 10 0 0 0 0 0 0 3 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 0 0 0 3 3 0 0 0 3 0 0 0 10 0 0 0 0 3 0 10 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 9
,
 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 1 9 3 3 12 0 0 0 0 4 0 9 0 0 0 0 12 0 0 0 4 0 0 3 1 9 0 3 12 0 0 0 2 0 0 0 1 0 0 0 0 5 0 9 0

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,10,3,6,0,0,0,0,3,10,0,10,3,0,0,3,0,10,0,10,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,3,0,10,0,9,0,0,0,3,10,0,0,0,9],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,1,0,12,1,2,0,0,0,9,4,0,9,0,5,0,0,3,0,0,0,0,0,0,0,3,9,0,3,0,9,0,0,12,0,4,12,1,0] >;

C3×C32⋊C12 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes C_{12}
% in TeX

G:=Group("C3xC3^2:C12");
// GroupNames label

G:=SmallGroup(324,92);
// by ID

G=gap.SmallGroup(324,92);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-3,108,2164,2170,7781]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
// generators/relations

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