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G = D6.1D12  order 288 = 25·32

1st non-split extension by D6 of D12 acting via D12/C12=C2

metabelian, supersoluble, monomial

Aliases: D6.1D12, C24.38D6, D12.19D6, Dic6.17D6, Dic3.12D12, C8.11S32, (S3×C8)⋊5S3, C3⋊C8.24D6, C6.8(S3×D4), (S3×C24)⋊3C2, C24⋊C2⋊7S3, C6.8(C2×D12), C24⋊2S3⋊8C2, C3⋊1(C4○D24), C3⋊D24⋊2C2, (S3×C6).20D4, (C4×S3).36D6, C2.13(S3×D12), C32⋊3(C4○D8), D12⋊5S3⋊2C2, D6.6D6⋊2C2, C32⋊3Q16⋊2C2, C3⋊1(Q8.7D6), (C3×C12).51C23, (C3×C24).36C22, C12.71(C22×S3), (C3×Dic3).25D4, (C3×D12).5C22, (S3×C12).44C22, C12⋊S3.4C22, (C3×Dic6).5C22, C32⋊4Q8.4C22, C4.47(C2×S32), (C3×C6).35(C2×D4), (C3×C24⋊C2)⋊11C2, (C3×C3⋊C8).29C22, SmallGroup(288,454)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D6.1D12
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — D12⋊5S3 — D6.1D12
Lower central
C32 — C3×C6 — C3×C12 — D6.1D12
Upper central
C1 — C2 — C4 — C8

Generators and relations for D6.1D12
 G = < a,b,c,d | a6=b2=1, c12=d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c11 >

Subgroups: 618 in 134 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, S3×C8, C24⋊C2, C24⋊C2, D24, Dic12, D4⋊S3, C3⋊Q16, C2×C24, C3×SD16, C4○D12, D4⋊2S3, Q8⋊3S3, C3×C3⋊C8, C3×C24, S3×Dic3, C6.D6, D6⋊S3, C3⋊D12, C3×Dic6, S3×C12, C3×D12, C32⋊4Q8, C12⋊S3, C4○D24, Q8.7D6, C3⋊D24, C32⋊3Q16, S3×C24, C3×C24⋊C2, C24⋊2S3, D12⋊5S3, D6.6D6, D6.1D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C4○D8, S32, C2×D12, S3×D4, C2×S32, C4○D24, Q8.7D6, S3×D12, D6.1D12

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

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

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

45 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F8A8B8C8D12A12B12C12D12E12F12G12H24A24B24C24D24E···24J24K24L24M24N
order1222233344444666666888812121212121212122424242424···2424242424
size116123622423312362246624226622444662422224···46666

45 irreducible representations

dim111111112222222222222444444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6D12D12C4○D8C4○D24S32S3×D4C2×S32Q8.7D6S3×D12D6.1D12
kernelD6.1D12C3⋊D24C32⋊3Q16S3×C24C3×C24⋊C2C24⋊2S3D12⋊5S3D6.6D6S3×C8C24⋊C2C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3D12Dic3D6C32C3C8C6C4C3C2C1
# reps111111111111121112248111224

Matrix representation of D6.1D12 ►in GL6(𝔽73)

7200000
0720000
0072100
0072000
000010
000001
,
5680000
37170000
0072000
0072100
000010
000001
,
26550000
8590000
0072000
0007200
0000721
0000720
,
4600000
13270000
001000
000100
0000720
0000721

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[56,37,0,0,0,0,8,17,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,8,0,0,0,0,55,59,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,13,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1] >;
 

D6.1D12 in GAP, Magma, Sage, TeX 

D_6._1D_{12}
 
% in TeX
 
G:=Group("D6.1D12");
 
// GroupNames label
 
G:=SmallGroup(288,454);
 
// by ID
 
G=gap.SmallGroup(288,454);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,135,58,346,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^2=1,c^12=d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^11>;
 
// generators/relations
 

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