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G = C3xC6.D6order 216 = 23·33

Direct product of C3 and C6.D6

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C3xC6.D6, C6.26S32, C3:S3:2C12, C3:1(S3xC12), C6.2(S3xC6), C33:5(C2xC4), C32:9(C4xS3), (C3xC6).39D6, C32:5(C2xC12), (C3xDic3):5S3, Dic3:2(C3xS3), (C3xDic3):3C6, (C32xDic3):4C2, (C32xC6).2C22, C2.2(C3xS32), (C3xC3:S3):1C4, (C2xC3:S3).2C6, (C6xC3:S3).1C2, (C3xC6).7(C2xC6), (C3xDic3)o(C3xDic3), SmallGroup(216,120)

Series: Derived Chief Lower central Upper central

C1C32 — C3xC6.D6
C1C3C32C3xC6C32xC6C32xDic3 — C3xC6.D6
C32 — C3xC6.D6
C1C6

Generators and relations for C3xC6.D6
 G = < a,b,c,d | a3=b6=d2=1, c6=b3, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c5 >

Subgroups: 268 in 90 conjugacy classes, 32 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C2xC4, C32, C32, C32, Dic3, C12, D6, C2xC6, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, C2xC12, C33, C3xDic3, C3xDic3, C3xC12, S3xC6, C2xC3:S3, C3xC3:S3, C32xC6, C6.D6, S3xC12, C32xDic3, C6xC3:S3, C3xC6.D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, C12, D6, C2xC6, C3xS3, C4xS3, C2xC12, S32, S3xC6, C6.D6, S3xC12, C3xS32, C3xC6.D6

Permutation representations of C3xC6.D6
On 24 points - transitive group 24T544
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 24)
(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 23)(2 16)(3 21)(4 14)(5 19)(6 24)(7 17)(8 22)(9 15)(10 20)(11 13)(12 18)

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

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

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

G:=TransitiveGroup(24,544);

C3xC6.D6 is a maximal subgroup of
C3:S3.2D12  C33:C4:C4  Dic3:6S32  C3:S3:4D12  C33:5(C2xQ8)  D6.S32  Dic3.S32  S32xC12
C3xC6.D6 is a maximal quotient of
C3xDic32

54 conjugacy classes

class 1 2A2B2C3A3B3C···3H3I3J3K4A4B4C4D6A6B6C···6H6I6J6K6L6M6N6O12A···12H12I···12T
order1222333···33334444666···6666666612···1212···12
size1199112···24443333112···244499993···36···6

54 irreducible representations

dim111111112222224444
type+++++++
imageC1C2C2C3C4C6C6C12S3D6C3xS3C4xS3S3xC6S3xC12S32C6.D6C3xS32C3xC6.D6
kernelC3xC6.D6C32xDic3C6xC3:S3C6.D6C3xC3:S3C3xDic3C2xC3:S3C3:S3C3xDic3C3xC6Dic3C32C6C3C6C3C2C1
# reps121244282244481122

Matrix representation of C3xC6.D6 in GL4(F7) generated by

4000
0400
0040
0004
,
1046
2410
6132
4451
,
3022
3316
2311
3140
,
2402
4223
1645
1636
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,2,6,4,0,4,1,4,4,1,3,5,6,0,2,1],[3,3,2,3,0,3,3,1,2,1,1,4,2,6,1,0],[2,4,1,1,4,2,6,6,0,2,4,3,2,3,5,6] >;

C3xC6.D6 in GAP, Magma, Sage, TeX

C_3\times C_6.D_6
% in TeX

G:=Group("C3xC6.D6");
// GroupNames label

G:=SmallGroup(216,120);
// by ID

G=gap.SmallGroup(216,120);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,79,730,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=d^2=1,c^6=b^3,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^5>;
// generators/relations

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