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G = D12.2Dic3  order 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, C3⋊5(C8○D12), C12.30(C4×S3), C4○D12.6S3, (C4×S3).31D6, (C3×D12).1C4, C32⋊4(C8○D4), C4.4(S3×Dic3), (C2×C12).107D6, C62.44(C2×C4), (C3×Dic6).1C4, D6.1(C2×Dic3), C3⋊D4.2Dic3, C12.58D6⋊8C2, D6.Dic3⋊10C2, (S3×C12).8C22, C3⋊1(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), C32⋊4C8.20C22, (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

Derived series
C1 — C3×C6 — D12.2Dic3
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — S3×C12 — S3×C3⋊C8 — D12.2Dic3
Lower central
C32 — C3×C6 — D12.2Dic3
Upper central
C1 — C4 — C2×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, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C8○D4, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C2×C24, C4○D12, C3×C4○D4, C3×C3⋊C8, C32⋊4C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C8○D12, D4.Dic3, S3×C3⋊C8, D6.Dic3, C6×C3⋊C8, C12.58D6, C3×C4○D12, D12.2Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C4×S3, C2×Dic3, C22×S3, C8○D4, S32, S3×C2×C4, C22×Dic3, S3×Dic3, 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 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)

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

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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