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G = D12:4Dic3order 288 = 25·32

4th semidirect product of D12 and Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D12:4Dic3, C62.23D4, Dic6:4Dic3, C32:3C4wrC2, (C3xD12):3C4, (C2xC6).8D12, C12.18(C4xS3), (C3xDic6):3C4, C4oD12.2S3, (C2xC12).79D6, C4.Dic3:4S3, C4.8(S3xDic3), C6.38(D6:C4), (C3xC12).109D4, C12.9(C2xDic3), C3:3(D12:C4), C12.95(C3:D4), C2.9(D6:Dic3), (C6xC12).30C22, C3:1(Q8:3Dic3), C4.17(D6:S3), C6.8(C6.D4), C22.7(C3:D12), (C2xC4).101S32, (C4xC3:Dic3):1C2, (C3xC12).28(C2xC4), (C3xC4oD12).1C2, (C2xC6).10(C3:D4), (C3xC4.Dic3):15C2, (C3xC6).35(C22:C4), SmallGroup(288,216)

Series: Derived Chief Lower central Upper central

C1C3xC12 — D12:4Dic3
C1C3C32C3xC6C3xC12C6xC12C3xC4oD12 — D12:4Dic3
C32C3xC6C3xC12 — D12:4Dic3
C1C4C2xC4

Generators and relations for D12:4Dic3
 G = < a,b,c,d | a12=b2=c6=1, d2=c3, bab=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 362 in 103 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, M4(2), C4oD4, C3xS3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C4wrC2, C3xDic3, C3:Dic3, C3xC12, S3xC6, C62, C4.Dic3, C4xDic3, C3xM4(2), C4oD12, C3xC4oD4, C3xC3:C8, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C2xC3:Dic3, C6xC12, D12:C4, Q8:3Dic3, C3xC4.Dic3, C4xC3:Dic3, C3xC4oD12, D12:4Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C4xS3, D12, C2xDic3, C3:D4, C4wrC2, S32, D6:C4, C6.D4, S3xDic3, D6:S3, C3:D12, D12:C4, Q8:3Dic3, D6:Dic3, D12:4Dic3

Permutation representations of D12:4Dic3
On 24 points - transitive group 24T612
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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 15 17 19 21 23)(14 16 18 20 22 24)
(2 6)(3 11)(5 9)(8 12)(13 16 19 22)(14 21 20 15)(17 24 23 18)

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

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

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

G:=TransitiveGroup(24,612);

42 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A6B6C···6G6H6I8A8B12A12B12C12D12E···12J12K12L24A24B24C24D
order122233344444444666···666881212121212···12121224242424
size112122241121218181818224···41212121222224···4121212121212

42 irreducible representations

dim1111112222222222224444444
type++++++++--+++--+
imageC1C2C2C2C4C4S3S3D4D4Dic3Dic3D6C4xS3C3:D4D12C3:D4C4wrC2S32S3xDic3D6:S3C3:D12D12:C4Q8:3Dic3D12:4Dic3
kernelD12:4Dic3C3xC4.Dic3C4xC3:Dic3C3xC4oD12C3xDic6C3xD12C4.Dic3C4oD12C3xC12C62Dic6D12C2xC12C12C12C2xC6C2xC6C32C2xC4C4C4C22C3C3C1
# reps1111221111112242241111224

Matrix representation of D12:4Dic3 in GL4(F5) generated by

0040
0001
4020
0103
,
0300
2000
0402
1030
,
1030
0003
3000
0304
,
2010
0103
0030
0004
G:=sub<GL(4,GF(5))| [0,0,4,0,0,0,0,1,4,0,2,0,0,1,0,3],[0,2,0,1,3,0,4,0,0,0,0,3,0,0,2,0],[1,0,3,0,0,0,0,3,3,0,0,0,0,3,0,4],[2,0,0,0,0,1,0,0,1,0,3,0,0,3,0,4] >;

D12:4Dic3 in GAP, Magma, Sage, TeX

D_{12}\rtimes_4{\rm Dic}_3
% in TeX

G:=Group("D12:4Dic3");
// GroupNames label

G:=SmallGroup(288,216);
// by ID

G=gap.SmallGroup(288,216);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,100,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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