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G = Dic3:5D12  order 288 = 25·32

2nd semidirect product of Dic3 and D12 acting through Inn(Dic3)

metabelian, supersoluble, monomial

Aliases: Dic3:5D12, C62.64C23, C12:6(C4xS3), C3:1(C4xD12), C32:6(C4xD4), C2.4(S3xD12), C6.18(S3xD4), C4:Dic3:17S3, (C3xDic3):9D4, (C4xDic3):6S3, C6.19(C2xD12), C12:S3:10C4, (C2xC12).281D6, C4:1(C6.D6), C3:1(Dic3:5D4), (Dic3xC12):11C2, C6.12(C4oD12), (C2xDic3).71D6, C6.D12:13C2, (C6xC12).102C22, C6.13(Q8:3S3), C2.4(D6.6D6), (C6xDic3).64C22, (C2xC4).80S32, C6.33(S3xC2xC4), (C3xC12):6(C2xC4), C22.36(C2xS32), (C3xC6).51(C2xD4), (C3xC4:Dic3):13C2, (C2xC6.D6):9C2, (C3xC6).38(C4oD4), (C3xC6).57(C22xC4), (C2xC6).83(C22xS3), C2.10(C2xC6.D6), (C2xC12:S3).13C2, (C22xC3:S3).17C22, (C2xC3:S3):2(C2xC4), SmallGroup(288,542)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — Dic3:5D12
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C6xDic3 — C2xC6.D6 — Dic3:5D12
Lower central
C32 — C3xC6 — Dic3:5D12
Upper central
C1 — C22 — C2xC4

Generators and relations for Dic3:5D12
 G = < a,b,c,d | a6=c12=d2=1, b2=a3, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

Subgroups: 962 in 215 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C4xD4, C3xDic3, C3xDic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, C4xDic3, C4:Dic3, D6:C4, C4xC12, C3xC4:C4, S3xC2xC4, C2xD12, C6.D6, C6xDic3, C6xDic3, C12:S3, C6xC12, C22xC3:S3, C4xD12, Dic3:5D4, C6.D12, Dic3xC12, C3xC4:Dic3, C2xC6.D6, C2xC12:S3, Dic3:5D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22xC4, C2xD4, C4oD4, C4xS3, D12, C22xS3, C4xD4, S32, S3xC2xC4, C2xD12, C4oD12, S3xD4, Q8:3S3, C6.D6, C2xS32, C4xD12, Dic3:5D4, D6.6D6, S3xD12, C2xC6.D6, Dic3:5D12

Smallest permutation representation of Dic3:5D12
►On 48 points
Generators in S48
(1 24 9 20 5 16)(2 13 10 21 6 17)(3 14 11 22 7 18)(4 15 12 23 8 19)(25 47 29 39 33 43)(26 48 30 40 34 44)(27 37 31 41 35 45)(28 38 32 42 36 46)

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F4G···4L6A···6F6G6H6I12A12B12C12D12E···12J12K···12R12S12T12U12V
order122222223334444444···46···66661212121212···1212···1212121212
size1111181818182242233336···62···244422224···46···612121212

54 irreducible representations

dim11111112222222224444444
type+++++++++++++++++++
imageC1C2C2C2C2C2C4S3S3D4D6D6C4oD4D12C4xS3C4oD12S32S3xD4Q8:3S3C6.D6C2xS32D6.6D6S3xD12
kernelDic3:5D12C6.D12Dic3xC12C3xC4:Dic3C2xC6.D6C2xC12:S3C12:S3C4xDic3C4:Dic3C3xDic3C2xDic3C2xC12C3xC6Dic3C12C6C2xC4C6C6C4C22C2C2
# reps12112181124224841112122

Matrix representation of Dic3:5D12 ►in GL6(F13)

010000
1210000
0012000
0001200
0000120
0000012
,
050000
500000
008000
000800
000050
000005
,
100000
010000
009300
003400
0000012
000011
,
010000
100000
001000
0071200
00001212
000001

G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,3,0,0,0,0,3,4,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,7,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1] >;
 

Dic3:5D12 in GAP, Magma, Sage, TeX 

{\rm Dic}_3\rtimes_5D_{12}
 
% in TeX
 
G:=Group("Dic3:5D12");
 
// GroupNames label
 
G:=SmallGroup(288,542);
 
// by ID
 
G=gap.SmallGroup(288,542);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=c^12=d^2=1,b^2=a^3,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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