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G = D12.8D6  order 288 = 25·32

8th non-split extension by D12 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: D12.8D6, Dic6.7D6, D4.9S32, D4⋊S3⋊5S3, C3⋊C8.14D6, D4.S3⋊5S3, C6.59(S3×D4), (C3×D4).13D6, D12⋊S3⋊4C2, C3⋊3(D8⋊3S3), C32⋊12(C4○D8), C3⋊Dic3.58D4, C32⋊3Q16⋊8C2, C12.29D6⋊1C2, C12.D6⋊2C2, C3⋊3(Q8.7D6), C12.13(C22×S3), (C3×C12).13C23, D12.S3⋊12C2, C2.19(Dic3⋊D6), (C3×D12).15C22, (D4×C32).9C22, C32⋊4Q8.9C22, (C3×Dic6).14C22, C4.13(C2×S32), (C3×D4⋊S3)⋊4C2, (C3×D4.S3)⋊8C2, (C2×C3⋊S3).23D4, (C3×C6).128(C2×D4), (C3×C3⋊C8).18C22, (C4×C3⋊S3).15C22, SmallGroup(288,584)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — D12.8D6
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C3×D12 — D12⋊S3 — D12.8D6
Lower central
C32 — C3×C6 — C3×C12 — D12.8D6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D12.8D6
 G = < a,b,c,d | a12=b2=c6=1, d2=a9, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c-1 >

Subgroups: 602 in 144 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C3×Q8, C4○D8, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, S3×C8, C24⋊C2, Dic12, D4⋊S3, D4.S3, D4.S3, C3⋊Q16, C3×D8, C3×SD16, D4⋊2S3, Q8⋊3S3, C3×C3⋊C8, S3×Dic3, C3⋊D12, C3×Dic6, C3×D12, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, D8⋊3S3, Q8.7D6, C12.29D6, D12.S3, C32⋊3Q16, C3×D4⋊S3, C3×D4.S3, D12⋊S3, C12.D6, D12.8D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C22×S3, C4○D8, S32, S3×D4, C2×S32, D8⋊3S3, Q8.7D6, Dic3⋊D6, D12.8D6

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

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

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

33 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B8C8D12A12B12C12D24A24B24C24D
order12222333444446666666688881212121224242424
size1141218224299123622488882466664482412121212

33 irreducible representations

dim111111112222222224444448
type+++++++++++++++++++-+-
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6C4○D8S32S3×D4C2×S32D8⋊3S3Q8.7D6Dic3⋊D6D12.8D6
kernelD12.8D6C12.29D6D12.S3C32⋊3Q16C3×D4⋊S3C3×D4.S3D12⋊S3C12.D6D4⋊S3D4.S3C3⋊Dic3C2×C3⋊S3C3⋊C8Dic6D12C3×D4C32D4C6C4C3C3C2C1
# reps111111111111211241212221

Matrix representation of D12.8D6 ►in GL6(𝔽73)

2700000
36460000
0072100
0072000
000010
000001
,
15140000
57580000
001000
0017200
0000720
0000072
,
4670000
39690000
0072000
0007200
0000721
0000720
,
6300000
3220000
001000
000100
000010
0000172

G:=sub<GL(6,GF(73))| [27,36,0,0,0,0,0,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,57,0,0,0,0,14,58,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[4,39,0,0,0,0,67,69,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[63,3,0,0,0,0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;
 

D12.8D6 in GAP, Magma, Sage, TeX 

D_{12}._8D_6
 
% in TeX
 
G:=Group("D12.8D6");
 
// GroupNames label
 
G:=SmallGroup(288,584);
 
// by ID
 
G=gap.SmallGroup(288,584);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,675,346,185,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^-1>;
 
// generators/relations
 

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