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G = D12.4D6order 288 = 25·32

4th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.4D6, C24.31D6, Dic12:6S3, Dic6.4D6, C8.3S32, C24:C2:4S3, C6.34(S3xD4), C24:S3:2C2, (C3xDic12):1C2, C3:2(D4.D6), C3:2(Q16:S3), C3:Dic3.14D4, (C3xC24).1C22, C32:2Q16:4C2, Dic3.D6:2C2, Dic6:S3:4C2, D12:S3.1C2, (C3xC12).56C23, C12.76(C22xS3), (C3xD12).9C22, C32:5(C8.C22), C2.11(D6:D6), C32:4C8.4C22, (C3xDic6).9C22, C4.71(C2xS32), (C3xC24:C2):2C2, (C2xC3:S3).18D4, (C3xC6).40(C2xD4), (C4xC3:S3).10C22, SmallGroup(288,459)

Series: Derived Chief Lower central Upper central

C1C3xC12 — D12.4D6
C1C3C32C3xC6C3xC12C3xD12D12:S3 — D12.4D6
C32C3xC6C3xC12 — D12.4D6
C1C2C4C8

Generators and relations for D12.4D6
 G = < a,b,c,d | a12=b2=1, c6=a3, d2=a6, bab=a-1, ac=ca, dad-1=a7, cbc-1=dbd-1=a3b, dcd-1=a9c5 >

Subgroups: 530 in 130 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3xS3, C3:S3, C3xC6, C3:C8, C24, C24, Dic6, Dic6, C4xS3, D12, D12, C2xDic3, C3:D4, C3xD4, C3xQ8, C8.C22, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C8:S3, C24:C2, Dic12, D4.S3, Q8:2S3, C3:Q16, C3xSD16, C3xQ16, D4:2S3, S3xQ8, Q8:3S3, C32:4C8, C3xC24, S3xDic3, C6.D6, C3:D12, C32:2Q8, C3xDic6, C3xD12, C4xC3:S3, D4.D6, Q16:S3, Dic6:S3, C32:2Q16, C3xC24:C2, C3xDic12, C24:S3, D12:S3, Dic3.D6, D12.4D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, C8.C22, S32, S3xD4, C2xS32, D4.D6, Q16:S3, D6:D6, D12.4D6

Smallest permutation representation of D12.4D6
On 48 points
Generators in S48
(1 3 5 7 9 11 13 15 17 19 21 23)(2 4 6 8 10 12 14 16 18 20 22 24)(25 35 45 31 41 27 37 47 33 43 29 39)(26 36 46 32 42 28 38 48 34 44 30 40)
(1 30)(2 25)(3 44)(4 39)(5 34)(6 29)(7 48)(8 43)(9 38)(10 33)(11 28)(12 47)(13 42)(14 37)(15 32)(16 27)(17 46)(18 41)(19 36)(20 31)(21 26)(22 45)(23 40)(24 35)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 27 13 39)(2 26 14 38)(3 25 15 37)(4 48 16 36)(5 47 17 35)(6 46 18 34)(7 45 19 33)(8 44 20 32)(9 43 21 31)(10 42 22 30)(11 41 23 29)(12 40 24 28)

G:=sub<Sym(48)| (1,3,5,7,9,11,13,15,17,19,21,23)(2,4,6,8,10,12,14,16,18,20,22,24)(25,35,45,31,41,27,37,47,33,43,29,39)(26,36,46,32,42,28,38,48,34,44,30,40), (1,30)(2,25)(3,44)(4,39)(5,34)(6,29)(7,48)(8,43)(9,38)(10,33)(11,28)(12,47)(13,42)(14,37)(15,32)(16,27)(17,46)(18,41)(19,36)(20,31)(21,26)(22,45)(23,40)(24,35), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,27,13,39)(2,26,14,38)(3,25,15,37)(4,48,16,36)(5,47,17,35)(6,46,18,34)(7,45,19,33)(8,44,20,32)(9,43,21,31)(10,42,22,30)(11,41,23,29)(12,40,24,28)>;

G:=Group( (1,3,5,7,9,11,13,15,17,19,21,23)(2,4,6,8,10,12,14,16,18,20,22,24)(25,35,45,31,41,27,37,47,33,43,29,39)(26,36,46,32,42,28,38,48,34,44,30,40), (1,30)(2,25)(3,44)(4,39)(5,34)(6,29)(7,48)(8,43)(9,38)(10,33)(11,28)(12,47)(13,42)(14,37)(15,32)(16,27)(17,46)(18,41)(19,36)(20,31)(21,26)(22,45)(23,40)(24,35), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,27,13,39)(2,26,14,38)(3,25,15,37)(4,48,16,36)(5,47,17,35)(6,46,18,34)(7,45,19,33)(8,44,20,32)(9,43,21,31)(10,42,22,30)(11,41,23,29)(12,40,24,28) );

G=PermutationGroup([[(1,3,5,7,9,11,13,15,17,19,21,23),(2,4,6,8,10,12,14,16,18,20,22,24),(25,35,45,31,41,27,37,47,33,43,29,39),(26,36,46,32,42,28,38,48,34,44,30,40)], [(1,30),(2,25),(3,44),(4,39),(5,34),(6,29),(7,48),(8,43),(9,38),(10,33),(11,28),(12,47),(13,42),(14,37),(15,32),(16,27),(17,46),(18,41),(19,36),(20,31),(21,26),(22,45),(23,40),(24,35)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,27,13,39),(2,26,14,38),(3,25,15,37),(4,48,16,36),(5,47,17,35),(6,46,18,34),(7,45,19,33),(8,44,20,32),(9,43,21,31),(10,42,22,30),(11,41,23,29),(12,40,24,28)]])

33 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D8A8B12A12B12C12D12E12F12G24A···24H
order1222333444446666881212121212121224···24
size1112182242121212182242443644442424244···4

33 irreducible representations

dim11111111222222244444444
type+++++++++++++++-+++-
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6C8.C22S32S3xD4C2xS32D4.D6Q16:S3D6:D6D12.4D6
kernelD12.4D6Dic6:S3C32:2Q16C3xC24:C2C3xDic12C24:S3D12:S3Dic3.D6C24:C2Dic12C3:Dic3C2xC3:S3C24Dic6D12C32C8C6C4C3C3C2C1
# reps11111111111123111212224

Matrix representation of D12.4D6 in GL4(F5) generated by

0020
1200
4441
4404
,
2332
1102
4343
1113
,
4401
4310
4142
2440
,
3302
0003
2321
0300
G:=sub<GL(4,GF(5))| [0,1,4,4,0,2,4,4,2,0,4,0,0,0,1,4],[2,1,4,1,3,1,3,1,3,0,4,1,2,2,3,3],[4,4,4,2,4,3,1,4,0,1,4,4,1,0,2,0],[3,0,2,0,3,0,3,3,0,0,2,0,2,3,1,0] >;

D12.4D6 in GAP, Magma, Sage, TeX

D_{12}._4D_6
% in TeX

G:=Group("D12.4D6");
// GroupNames label

G:=SmallGroup(288,459);
// by ID

G=gap.SmallGroup(288,459);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,303,58,675,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^3,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^9*c^5>;
// generators/relations

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