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G = Dic6.29D6order 288 = 25·32

3rd non-split extension by Dic6 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: C12.25D12, C62.52D4, Dic6.29D6, C3:C8.4D6, (C2xDic6):9S3, (C6xDic6):4C2, (C2xC6).62D12, C6.72(C2xD12), (C3xC12).70D4, (C2xC12).122D6, C3:4(C8.D6), C4.Dic3:10S3, C32:3Q16:6C2, C32:5SD16:4C2, C12.49(C3:D4), (C6xC12).82C22, (C3xC12).69C23, C12.85(C22xS3), C32:8(C8.C22), C4.24(C3:D12), C3:1(Q8.11D6), C12.59D6.5C2, C12:S3.28C22, C22.5(C3:D12), (C3xDic6).36C22, C32:4Q8.28C22, C4.57(C2xS32), (C2xC4).10S32, C6.8(C2xC3:D4), (C3xC6).73(C2xD4), (C3xC3:C8).4C22, (C3xC4.Dic3):4C2, C2.12(C2xC3:D12), (C2xC6).21(C3:D4), SmallGroup(288,481)

Series: Derived Chief Lower central Upper central

C1C3xC12 — Dic6.29D6
C1C3C32C3xC6C3xC12C3xDic6C32:5SD16 — Dic6.29D6
C32C3xC6C3xC12 — Dic6.29D6
C1C2C2xC4

Generators and relations for Dic6.29D6
 G = < a,b,c,d | a12=c6=1, b2=a6, d2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >

Subgroups: 570 in 140 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, M4(2), SD16, Q16, C2xQ8, C4oD4, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C3xQ8, C8.C22, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C62, C24:C2, Dic12, C4.Dic3, Q8:2S3, C3:Q16, C3xM4(2), C2xDic6, C4oD12, C6xQ8, C3xC3:C8, C3xDic6, C3xDic6, C6xDic3, C32:4Q8, C4xC3:S3, C12:S3, C32:7D4, C6xC12, C8.D6, Q8.11D6, C32:5SD16, C32:3Q16, C3xC4.Dic3, C6xDic6, C12.59D6, Dic6.29D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C3:D4, C22xS3, C8.C22, S32, C2xD12, C2xC3:D4, C3:D12, C2xS32, C8.D6, Q8.11D6, C2xC3:D12, Dic6.29D6

Smallest permutation representation of Dic6.29D6
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 33 7 27)(2 32 8 26)(3 31 9 25)(4 30 10 36)(5 29 11 35)(6 28 12 34)(13 42 19 48)(14 41 20 47)(15 40 21 46)(16 39 22 45)(17 38 23 44)(18 37 24 43)
(1 15 9 23 5 19)(2 16 10 24 6 20)(3 17 11 13 7 21)(4 18 12 14 8 22)(25 44 29 48 33 40)(26 45 30 37 34 41)(27 46 31 38 35 42)(28 47 32 39 36 43)
(1 45 4 48 7 39 10 42)(2 46 5 37 8 40 11 43)(3 47 6 38 9 41 12 44)(13 30 16 33 19 36 22 27)(14 31 17 34 20 25 23 28)(15 32 18 35 21 26 24 29)

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

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

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

39 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G6H8A8B12A12B12C···12I12J12K12L12M24A24B24C24D
order1222333444446666666688121212···121212121224242424
size1123622422121236222244441212224···41212121212121212

39 irreducible representations

dim1111112222222222244444444
type+++++++++++++++-++++-
imageC1C2C2C2C2C2S3S3D4D4D6D6D6D12C3:D4D12C3:D4C8.C22S32C3:D12C2xS32C3:D12C8.D6Q8.11D6Dic6.29D6
kernelDic6.29D6C32:5SD16C32:3Q16C3xC4.Dic3C6xDic6C12.59D6C4.Dic3C2xDic6C3xC12C62C3:C8Dic6C2xC12C12C12C2xC6C2xC6C32C2xC4C4C4C22C3C3C1
# reps1221111111222222211111224

Matrix representation of Dic6.29D6 in GL4(F73) generated by

14700
66700
00147
00667
,
00720
0011
1000
727200
,
134300
304300
004330
004313
,
003043
003060
02700
464600
G:=sub<GL(4,GF(73))| [14,66,0,0,7,7,0,0,0,0,14,66,0,0,7,7],[0,0,1,72,0,0,0,72,72,1,0,0,0,1,0,0],[13,30,0,0,43,43,0,0,0,0,43,43,0,0,30,13],[0,0,0,46,0,0,27,46,30,30,0,0,43,60,0,0] >;

Dic6.29D6 in GAP, Magma, Sage, TeX

{\rm Dic}_6._{29}D_6
% in TeX

G:=Group("Dic6.29D6");
// GroupNames label

G:=SmallGroup(288,481);
// by ID

G=gap.SmallGroup(288,481);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,64,219,675,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^6=1,b^2=a^6,d^2=a^3,b*a*b^-1=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=a^6*c^-1>;
// generators/relations

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