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## G = C3×C3⋊D12order 216 = 23·33

### Direct product of C3 and C3⋊D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C3⋊D12
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×C3⋊D12
 Lower central C32 — C3×C6 — C3×C3⋊D12
 Upper central C1 — C6

Generators and relations for C3×C3⋊D12
G = < a,b,c,d | a3=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 316 in 94 conjugacy classes, 28 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, D4, C32, C32, Dic3, C12, D6, D6, C2×C6, C3×S3, C3⋊S3, C3×C6, C3×C6, D12, C3⋊D4, C3×D4, C33, C3×Dic3, C3×Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C62, S3×C32, C3×C3⋊S3, C32×C6, C3⋊D12, C3×D12, C3×C3⋊D4, C32×Dic3, S3×C3×C6, C6×C3⋊S3, C3×C3⋊D12
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, D12, C3⋊D4, C3×D4, S32, S3×C6, C3⋊D12, C3×D12, C3×C3⋊D4, C3×S32, C3×C3⋊D12

Permutation representations of C3×C3⋊D12
On 24 points - transitive group 24T543
Generators in S24
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)
(1 9 5)(2 6 10)(3 11 7)(4 8 12)(13 17 21)(14 22 18)(15 19 23)(16 24 20)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)

G:=sub<Sym(24)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)>;

G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24), (1,9,5)(2,6,10)(3,11,7)(4,8,12)(13,17,21)(14,22,18)(15,19,23)(16,24,20), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19) );

G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24)], [(1,9,5),(2,6,10),(3,11,7),(4,8,12),(13,17,21),(14,22,18),(15,19,23),(16,24,20)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19)]])

G:=TransitiveGroup(24,543);

C3×C3⋊D12 is a maximal subgroup of
D6⋊S32  (S3×C6)⋊D6  C3⋊S34D12  D6.S32  D6.4S32  D6.3S32  D6.6S32  C3×S3×D12  C3×S3×C3⋊D4

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4 6A 6B 6C ··· 6H 6I 6J 6K 6L ··· 6S 6T 6U 12A ··· 12H order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 6 6 6 ··· 6 6 6 6 6 ··· 6 6 6 12 ··· 12 size 1 1 6 18 1 1 2 ··· 2 4 4 4 6 1 1 2 ··· 2 4 4 4 6 ··· 6 18 18 6 ··· 6

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 S3 D4 D6 C3×S3 C3×S3 D12 C3⋊D4 C3×D4 S3×C6 C3×D12 C3×C3⋊D4 S32 C3⋊D12 C3×S32 C3×C3⋊D12 kernel C3×C3⋊D12 C32×Dic3 S3×C3×C6 C6×C3⋊S3 C3⋊D12 C3×Dic3 S3×C6 C2×C3⋊S3 C3×Dic3 S3×C6 C33 C3×C6 Dic3 D6 C32 C32 C32 C6 C3 C3 C6 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 2 2 2 2 2 4 4 4 1 1 2 2

Matrix representation of C3×C3⋊D12 in GL4(𝔽7) generated by

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

C3×C3⋊D12 in GAP, Magma, Sage, TeX

C_3\times C_3\rtimes D_{12}
% in TeX

G:=Group("C3xC3:D12");
// GroupNames label

G:=SmallGroup(216,122);
// by ID

G=gap.SmallGroup(216,122);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,79,730,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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