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G = D12.2Dic3order 288 = 25·32

The non-split extension by D12 of Dic3 acting through Inn(D12)

metabelian, supersoluble, monomial

Aliases: D12.2Dic3, Dic6.2Dic3, C3⋊C8.31D6, C35(C8○D12), C12.30(C4×S3), C4○D12.6S3, (C4×S3).31D6, (C3×D12).1C4, C324(C8○D4), C4.4(S3×Dic3), (C2×C12).107D6, C62.44(C2×C4), (C3×Dic6).1C4, D6.1(C2×Dic3), C3⋊D4.2Dic3, C12.58D68C2, D6.Dic310C2, (S3×C12).8C22, C31(D4.Dic3), (C6×C12).66C22, C12.25(C2×Dic3), C22.1(S3×Dic3), C6.3(C22×Dic3), (C3×C12).143C23, C12.142(C22×S3), Dic3.1(C2×Dic3), C324C8.20C22, (C2×C3⋊C8)⋊3S3, (S3×C3⋊C8)⋊9C2, (C6×C3⋊C8)⋊16C2, C4.89(C2×S32), (C2×C4).62S32, C6.83(S3×C2×C4), C2.5(C2×S3×Dic3), (S3×C6).4(C2×C4), (C2×C6).16(C4×S3), (C3×C3⋊D4).1C4, (C3×C12).56(C2×C4), (C3×C4○D12).5C2, (C3×C3⋊C8).38C22, (C3×C6).39(C22×C4), (C3×Dic3).4(C2×C4), (C2×C6).18(C2×Dic3), SmallGroup(288,462)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D12.2Dic3
C1C3C32C3×C6C3×C12S3×C12S3×C3⋊C8 — D12.2Dic3
C32C3×C6 — D12.2Dic3
C1C4C2×C4

Generators and relations for D12.2Dic3
 G = < a,b,c,d | a12=b2=1, c6=a6, d2=a6c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 338 in 134 conjugacy classes, 60 normal (38 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6 [×2], C6 [×6], C8 [×4], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], C2×C6 [×2], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], C4○D4, C3×S3 [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×6], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×3], C3×D4 [×3], C3×Q8, C8○D4, C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C2×C3⋊C8 [×2], C4.Dic3 [×5], C2×C24, C4○D12, C3×C4○D4, C3×C3⋊C8 [×2], C324C8 [×2], C3×Dic6, S3×C12 [×2], C3×D12, C3×C3⋊D4 [×2], C6×C12, C8○D12, D4.Dic3, S3×C3⋊C8 [×2], D6.Dic3 [×2], C6×C3⋊C8, C12.58D6, C3×C4○D12, D12.2Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3 [×2], C2×S32, C8○D12, D4.Dic3, C2×S3×Dic3, D12.2Dic3

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

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

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

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

54 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H6I6J8A8B8C8D8E8F8G8H8I8J12A···12F12G···12K12L12M24A···24H
order12222333444446666666666888888888812···1212···12121224···24
size1126622411266222244441212333366181818182···24···412126···6

54 irreducible representations

dim111111111222222222222444444
type+++++++++-+--++-+-
imageC1C2C2C2C2C2C4C4C4S3S3D6Dic3D6Dic3Dic3D6C4×S3C4×S3C8○D4C8○D12S32S3×Dic3C2×S32S3×Dic3D4.Dic3D12.2Dic3
kernelD12.2Dic3S3×C3⋊C8D6.Dic3C6×C3⋊C8C12.58D6C3×C4○D12C3×Dic6C3×D12C3×C3⋊D4C2×C3⋊C8C4○D12C3⋊C8Dic6C4×S3D12C3⋊D4C2×C12C12C2×C6C32C3C2×C4C4C4C22C3C1
# reps122111224112121222248111124

Matrix representation of D12.2Dic3 in GL6(𝔽73)

1720000
100000
0046000
00462700
000010
000001
,
100000
1720000
00271900
00274600
0000720
0000072
,
7200000
0720000
0046000
0004600
000001
0000721
,
2700000
0270000
0022000
0002200
0000046
0000460

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,46,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,27,27,0,0,0,0,19,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;

D12.2Dic3 in GAP, Magma, Sage, TeX

D_{12}._2{\rm Dic}_3
% in TeX

G:=Group("D12.2Dic3");
// GroupNames label

G:=SmallGroup(288,462);
// by ID

G=gap.SmallGroup(288,462);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,80,1356,9414]);
// 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,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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