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G = C3×C3⋊D12order 216 = 23·33

Direct product of C3 and C3⋊D12

direct product, metabelian, supersoluble, monomial

Aliases: C3×C3⋊D12, C335D4, C329D12, C6.28S32, (S3×C6)⋊2C6, (S3×C6)⋊2S3, D62(C3×S3), C32(C3×D12), C6.4(S3×C6), Dic3⋊(C3×S3), (C3×C6).41D6, C325(C3×D4), (C3×Dic3)⋊4S3, (C3×Dic3)⋊1C6, C329(C3⋊D4), (C32×Dic3)⋊2C2, (C32×C6).4C22, (S3×C3×C6)⋊2C2, C2.4(C3×S32), (C6×C3⋊S3)⋊1C2, (C2×C3⋊S3)⋊4C6, C31(C3×C3⋊D4), (C3×C6).9(C2×C6), SmallGroup(216,122)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C3×C3⋊D12
C1C3C32C3×C6C32×C6S3×C3×C6 — C3×C3⋊D12
C32C3×C6 — C3×C3⋊D12
C1C6

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 [×2], C3 [×3], C3 [×4], C4, C22 [×2], S3 [×5], C6 [×3], C6 [×9], D4, C32 [×3], C32 [×4], Dic3, C12 [×3], D6, D6 [×3], C2×C6 [×4], C3×S3 [×8], C3⋊S3, C3×C6 [×3], C3×C6 [×5], D12, C3⋊D4, C3×D4, C33, C3×Dic3 [×2], C3×Dic3, C3×C12, S3×C6 [×2], S3×C6 [×4], 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 [×3], C3, C22, S3 [×2], C6 [×3], D4, D6 [×2], C2×C6, C3×S3 [×2], D12, C3⋊D4, C3×D4, S32, S3×C6 [×2], 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 2A2B2C3A3B3C···3H3I3J3K 4 6A6B6C···6H6I6J6K6L···6S6T6U12A···12H
order1222333···33334666···66666···66612···12
size11618112···24446112···24446···618186···6

45 irreducible representations

dim111111112222222222224444
type+++++++++++
imageC1C2C2C2C3C6C6C6S3S3D4D6C3×S3C3×S3D12C3⋊D4C3×D4S3×C6C3×D12C3×C3⋊D4S32C3⋊D12C3×S32C3×C3⋊D12
kernelC3×C3⋊D12C32×Dic3S3×C3×C6C6×C3⋊S3C3⋊D12C3×Dic3S3×C6C2×C3⋊S3C3×Dic3S3×C6C33C3×C6Dic3D6C32C32C32C6C3C3C6C3C2C1
# reps111122221112222224441122

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

4000
0400
0040
0004
,
2656
4313
1125
1635
,
0515
4126
5210
5005
,
6124
1465
4043
3140
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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