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G = C24.64D6order 288 = 25·32

17th non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C24.64D6, C8.12S32, (S3xC8):8S3, C3:C8.30D6, C8:S3:7S3, D6.3(C4xS3), (S3xC24):14C2, C24:S3:8C2, C3:2(C8oD12), (C4xS3).29D6, C32:2(C8oD4), C3:1(D12.C4), C3:D12.1C4, D6:S3.1C4, Dic3.1(C4xS3), C32:2Q8.1C4, C12.29D6:9C2, D6.Dic3:12C2, (C3xC24).46C22, D6.D6.2C2, (S3xC12).39C22, (C3xC12).139C23, C12.138(C22xS3), C32:4C8.19C22, (S3xC3:C8):8C2, C2.7(C4xS32), C6.5(S3xC2xC4), C4.85(C2xS32), (S3xC6).2(C2xC4), (C3xC8:S3):11C2, (C3xC3:C8).24C22, (C3xC6).5(C22xC4), (C4xC3:S3).58C22, C3:Dic3.18(C2xC4), (C3xDic3).12(C2xC4), (C2xC3:S3).14(C2xC4), SmallGroup(288,452)

Series: Derived Chief Lower central Upper central

C1C3xC6 — C24.64D6
C1C3C32C3xC6C3xC12S3xC12D6.D6 — C24.64D6
C32C3xC6 — C24.64D6
C1C4C8

Generators and relations for C24.64D6
 G = < a,b,c | a24=b6=1, c2=a12, bab-1=cac-1=a5, cbc-1=a12b-1 >

Subgroups: 434 in 134 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2xC4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC8, M4(2), C4oD4, C3xS3, C3:S3, C3xC6, C3:C8, C3:C8, C24, C24, Dic6, C4xS3, C4xS3, D12, C3:D4, C2xC12, C8oD4, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, S3xC8, S3xC8, C8:S3, C8:S3, C2xC3:C8, C4.Dic3, C2xC24, C3xM4(2), C4oD12, C3xC3:C8, C32:4C8, C3xC24, D6:S3, C3:D12, C32:2Q8, S3xC12, C4xC3:S3, C8oD12, D12.C4, S3xC3:C8, C12.29D6, D6.Dic3, S3xC24, C3xC8:S3, C24:S3, D6.D6, C24.64D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, C8oD4, S32, S3xC2xC4, C2xS32, C8oD12, D12.C4, C4xS32, C24.64D6

Smallest permutation representation of C24.64D6
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 35 17 43 9 27)(2 40 18 48 10 32)(3 45 19 29 11 37)(4 26 20 34 12 42)(5 31 21 39 13 47)(6 36 22 44 14 28)(7 41 23 25 15 33)(8 46 24 30 16 38)
(1 23 13 11)(2 4 14 16)(3 9 15 21)(5 19 17 7)(6 24 18 12)(8 10 20 22)(25 27 37 39)(26 32 38 44)(28 42 40 30)(29 47 41 35)(31 33 43 45)(34 48 46 36)

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

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

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

54 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E12F12G12H12I24A24B24C24D24E···24J24K24L24M24N24O24P
order122223334444466666688888888881212121212121212122424242424···24242424242424
size1166182241166182246612223333661818222244661222224···466661212

54 irreducible representations

dim1111111111122222222244444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4S3S3D6D6D6C4xS3C4xS3C8oD4C8oD12S32C2xS32D12.C4C4xS32C24.64D6
kernelC24.64D6S3xC3:C8C12.29D6D6.Dic3S3xC24C3xC8:S3C24:S3D6.D6D6:S3C3:D12C32:2Q8S3xC8C8:S3C3:C8C24C4xS3Dic3D6C32C3C8C4C3C2C1
# reps1111111124211222444811224

Matrix representation of C24.64D6 in GL4(F5) generated by

3311
3232
2111
1231
,
4222
3122
4442
0231
,
1321
0443
2121
4323
G:=sub<GL(4,GF(5))| [3,3,2,1,3,2,1,2,1,3,1,3,1,2,1,1],[4,3,4,0,2,1,4,2,2,2,4,3,2,2,2,1],[1,0,2,4,3,4,1,3,2,4,2,2,1,3,1,3] >;

C24.64D6 in GAP, Magma, Sage, TeX

C_{24}._{64}D_6
% in TeX

G:=Group("C24.64D6");
// GroupNames label

G:=SmallGroup(288,452);
// by ID

G=gap.SmallGroup(288,452);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,219,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^12*b^-1>;
// generators/relations

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