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G = D6.D12  order 288 = 25·32

5th non-split extension by D6 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D6.8D12, C62.60C23, D6:C4:16S3, C4:Dic3:6S3, (S3xC6).6D4, C6.14(S3xD4), C6.15(C2xD12), C2.18(S3xD12), (C2xC12).228D6, Dic3:Dic3:5C2, C6.54(C4oD12), C3:4(D6.D4), C6.D12:7C2, (C2xDic3).24D6, (C22xS3).37D6, C6.11D12:10C2, C6.41(D4:2S3), (C6xC12).184C22, C6.34(Q8:3S3), C2.17(D12:S3), C2.17(D6.3D6), C3:1(C23.21D6), (C6xDic3).13C22, C32:6(C22.D4), (C2xC4).25S32, (C3xD6:C4):14C2, (C2xS3xDic3):12C2, (C3xC6).47(C2xD4), (C3xC4:Dic3):12C2, C22.106(C2xS32), (S3xC2xC6).19C22, (C2xC3:D12).6C2, (C3xC6).36(C4oD4), (C2xC6).79(C22xS3), (C22xC3:S3).16C22, (C2xC3:Dic3).44C22, SmallGroup(288,538)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — D6.D12
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xS3xDic3 — D6.D12
Lower central
C32 — C62 — D6.D12
Upper central
C1 — C22 — C2xC4

Generators and relations for D6.D12
 G = < a,b,c,d | a6=b2=c12=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c-1 >

Subgroups: 762 in 173 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C22.D4, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C2xC3:S3, C62, Dic3:C4, C4:Dic3, C4:Dic3, D6:C4, D6:C4, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, S3xDic3, C3:D12, C6xDic3, C2xC3:Dic3, C6xC12, S3xC2xC6, C22xC3:S3, C23.21D6, D6.D4, C6.D12, Dic3:Dic3, C3xC4:Dic3, C3xD6:C4, C6.11D12, C2xS3xDic3, C2xC3:D12, D6.D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C22xS3, C22.D4, S32, C2xD12, C4oD12, S3xD4, D4:2S3, Q8:3S3, C2xS32, C23.21D6, D6.D4, D12:S3, S3xD12, D6.3D6, D6.D12

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G6H6I6J6K12A···12H12I···12N
order122222233344444446···66666612···1212···12
size11116636224466121218182···244412124···412···12

42 irreducible representations

dim1111111122222222244444444
type+++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4oD4D12C4oD12S32S3xD4D4:2S3Q8:3S3C2xS32D12:S3S3xD12D6.3D6
kernelD6.D12C6.D12Dic3:Dic3C3xC4:Dic3C3xD6:C4C6.11D12C2xS3xDic3C2xC3:D12C4:Dic3D6:C4S3xC6C2xDic3C2xC12C22xS3C3xC6D6C6C2xC4C6C6C6C22C2C2C2
# reps1111111111232144411211222

Matrix representation of D6.D12 ►in GL8(F13)

120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
512000000
118000000
00100000
00010000
00001000
000011200
00000010
00000001
,
80000000
25000000
001220000
001210000
000012000
000001200
000000121
000000120
,
81000000
05000000
001200000
001210000
000012000
000001200
000000120
000000121

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,11,0,0,0,0,0,0,12,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,2,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[8,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1] >;
 

D6.D12 in GAP, Magma, Sage, TeX 

D_6.D_{12}
 
% in TeX
 
G:=Group("D6.D12");
 
// GroupNames label
 
G:=SmallGroup(288,538);
 
// by ID
 
G=gap.SmallGroup(288,538);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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