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G = D12:D6order 288 = 25·32

4th semidirect product of D12 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12:4D6, D4:3S32, C3:S3:3D8, C3:C8:13D6, C3:3(S3xD8), D4:S3:2S3, (C3xD4):1D6, C32:8(C2xD8), C6.55(S3xD4), D6:D6:3C2, C3:D24:8C2, (C3xD12):6C22, C3:Dic3.19D4, (C3xC12).3C23, C12.3(C22xS3), C12:S3:4C22, C12.29D6:3C2, (D4xC32):3C22, C2.15(Dic3:D6), C4.3(C2xS32), (D4xC3:S3):1C2, (C3xD4:S3):2C2, (C3xC3:C8):6C22, (C2xC3:S3).56D4, (C3xC6).118(C2xD4), (C4xC3:S3).11C22, SmallGroup(288,574)

Series: Derived Chief Lower central Upper central

C1C3xC12 — D12:D6
C1C3C32C3xC6C3xC12C3xD12D6:D6 — D12:D6
C32C3xC6C3xC12 — D12:D6
C1C2C4D4

Generators and relations for D12:D6
 G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=a7b, dcd=c-1 >

Subgroups: 1026 in 179 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, D4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC8, D8, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C3:C8, C24, C4xS3, D12, D12, C3:D4, C3xD4, C3xD4, C22xS3, C2xD8, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, S3xC8, D24, D4:S3, D4:S3, C3xD8, S3xD4, C3xC3:C8, D6:S3, C3xD12, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C2xS32, C22xC3:S3, S3xD8, C12.29D6, C3:D24, C3xD4:S3, D6:D6, D4xC3:S3, D12:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S32, S3xD4, C2xS32, S3xD8, Dic3:D6, D12:D6

Permutation representations of D12:D6
On 24 points - transitive group 24T668
Generators in S24
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 14 21 22 17 18)(15 16 23 24 19 20)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)

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

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

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

G:=TransitiveGroup(24,668);

33 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D6E6F6G6H6I8A8B8C8D12A12B12C24A24B24C24D
order1222222233344666666666888812121224242424
size1149912123622421822488882424666644812121212

33 irreducible representations

dim1111112222222444448
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6D8S32S3xD4C2xS32S3xD8Dic3:D6D12:D6
kernelD12:D6C12.29D6C3:D24C3xD4:S3D6:D6D4xC3:S3D4:S3C3:Dic3C2xC3:S3C3:C8D12C3xD4C3:S3D4C6C4C3C2C1
# reps1122112112224121421

Matrix representation of D12:D6 in GL6(F73)

010000
7200000
0072100
0072000
000010
000001
,
16160000
16570000
001000
0017200
000010
000001
,
100000
0720000
0072000
0007200
000001
00007272
,
100000
0720000
0007200
0072000
000001
000010

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D12:D6 in GAP, Magma, Sage, TeX

D_{12}\rtimes D_6
% in TeX

G:=Group("D12:D6");
// GroupNames label

G:=SmallGroup(288,574);
// by ID

G=gap.SmallGroup(288,574);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,675,346,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations

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