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G = D12.Dic3order 288 = 25·32

The non-split extension by D12 of Dic3 acting via Dic3/C6=C2

metabelian, supersoluble, monomial

Aliases: D12.Dic3, Dic6.Dic3, C3:C8.21D6, C3:D4.Dic3, C12.31(C4xS3), C4oD12.3S3, (C4xS3).32D6, (C3xD12).2C4, C32:5(C8oD4), C4.Dic3:5S3, C4.9(S3xDic3), C3:5(D12.C4), (C2xC12).108D6, C62.45(C2xC4), (C3xDic6).2C4, D6.2(C2xDic3), D6.Dic3:14C2, (S3xC12).9C22, C3:2(D4.Dic3), (C6xC12).67C22, C12.17(C2xDic3), C22.2(S3xDic3), C6.4(C22xDic3), (C3xC12).144C23, C12.143(C22xS3), Dic3.2(C2xDic3), C32:4C8.38C22, (S3xC3:C8):13C2, C4.90(C2xS32), C6.84(S3xC2xC4), (C2xC4).105S32, C2.6(C2xS3xDic3), (S3xC6).5(C2xC4), (C2xC6).17(C4xS3), (C3xC3:D4).2C4, (C3xC12).57(C2xC4), (C2xC32:4C8):4C2, (C3xC4oD12).6C2, (C3xC3:C8).26C22, (C2xC6).8(C2xDic3), (C3xC4.Dic3):10C2, (C3xC6).40(C22xC4), (C3xDic3).5(C2xC4), SmallGroup(288,463)

Series: Derived Chief Lower central Upper central

C1C3xC6 — D12.Dic3
C1C3C32C3xC6C3xC12S3xC12S3xC3:C8 — D12.Dic3
C32C3xC6 — D12.Dic3
C1C4C2xC4

Generators and relations for D12.Dic3
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a6c3, bab=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a6b, dcd-1=c5 >

Subgroups: 338 in 135 conjugacy classes, 60 normal (36 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, C2xC8, M4(2), C4oD4, C3xS3, C3xC6, C3xC6, C3:C8, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C8oD4, C3xDic3, C3xC12, S3xC6, C62, S3xC8, C8:S3, C2xC3:C8, C4.Dic3, C4.Dic3, C3xM4(2), C4oD12, C3xC4oD4, C3xC3:C8, C32:4C8, C3xDic6, S3xC12, C3xD12, C3xC3:D4, C6xC12, D12.C4, D4.Dic3, S3xC3:C8, D6.Dic3, C3xC4.Dic3, C2xC32:4C8, C3xC4oD12, D12.Dic3
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, Dic3, D6, C22xC4, C4xS3, C2xDic3, C22xS3, C8oD4, S32, S3xC2xC4, C22xDic3, S3xDic3, C2xS32, D12.C4, D4.Dic3, C2xS3xDic3, D12.Dic3

Smallest permutation representation of D12.Dic3
On 48 points
Generators in S48
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 36)(12 35)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)
(1 6 11 4 9 2 7 12 5 10 3 8)(13 20 15 22 17 24 19 14 21 16 23 18)(25 32 27 34 29 36 31 26 33 28 35 30)(37 42 47 40 45 38 43 48 41 46 39 44)
(1 22 10 13 7 16 4 19)(2 17 11 20 8 23 5 14)(3 24 12 15 9 18 6 21)(25 37 28 46 31 43 34 40)(26 44 29 41 32 38 35 47)(27 39 30 48 33 45 36 42)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C···6G6H6I8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E···12J12K12L24A24B24C24D
order1222233344444666···66688888888881212121212···12121224242424
size1126622411266224···4121266669999181822224···4121212121212

48 irreducible representations

dim111111111222222222224444444
type+++++++++-+--++-+-
imageC1C2C2C2C2C2C4C4C4S3S3D6Dic3D6Dic3Dic3D6C4xS3C4xS3C8oD4S32S3xDic3C2xS32S3xDic3D12.C4D4.Dic3D12.Dic3
kernelD12.Dic3S3xC3:C8D6.Dic3C3xC4.Dic3C2xC32:4C8C3xC4oD12C3xDic6C3xD12C3xC3:D4C4.Dic3C4oD12C3:C8Dic6C4xS3D12C3:D4C2xC12C12C2xC6C32C2xC4C4C4C22C3C3C1
# reps122111224112121222241111224

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

0114
2323
2201
1122
,
2402
2310
0421
2033
,
4010
0203
2040
0101
,
0304
1020
0203
1010
G:=sub<GL(4,GF(5))| [0,2,2,1,1,3,2,1,1,2,0,2,4,3,1,2],[2,2,0,2,4,3,4,0,0,1,2,3,2,0,1,3],[4,0,2,0,0,2,0,1,1,0,4,0,0,3,0,1],[0,1,0,1,3,0,2,0,0,2,0,1,4,0,3,0] >;

D12.Dic3 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,463);
// by ID

G=gap.SmallGroup(288,463);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,219,80,1356,9414]);
// Polycyclic

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

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