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G = C24.D6order 288 = 25·32

46th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.46D6, C8.19S32, C3:C8.20D6, C8:S3:6S3, D6.1(C4xS3), (C4xS3).30D6, C32:3(C8oD4), C3:2(D12.C4), C3:D12.2C4, D6:S3.2C4, Dic3.2(C4xS3), C32:2Q8.2C4, (S3xC12).7C22, (C3xC24).47C22, D6.D6.3C2, C12.31D6:10C2, C12.139(C22xS3), (C3xC12).140C23, C32:4C8.37C22, C2.8(C4xS32), C6.6(S3xC2xC4), (S3xC3:C8):11C2, C4.86(C2xS32), (C8xC3:S3):14C2, (S3xC6).3(C2xC4), (C3xC8:S3):12C2, (C3xC3:C8).25C22, (C3xC6).6(C22xC4), (C4xC3:S3).87C22, C3:Dic3.32(C2xC4), (C3xDic3).3(C2xC4), (C2xC3:S3).28(C2xC4), SmallGroup(288,453)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C24.D6
 G = < a,b,c | a24=b6=1, c2=a12, bab-1=a5, cac-1=a17, cbc-1=a12b-1 >

Subgroups: 434 in 135 conjugacy classes, 50 normal (18 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, C8:S3, C8:S3, C2xC3:C8, C3xM4(2), C4oD12, C3xC3:C8, C32:4C8, C3xC24, D6:S3, C3:D12, C32:2Q8, S3xC12, C4xC3:S3, D12.C4, S3xC3:C8, C12.31D6, C3xC8:S3, C8xC3:S3, D6.D6, C24.D6
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, C8oD4, S32, S3xC2xC4, C2xS32, D12.C4, C4xS32, C24.D6

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

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

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

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

48 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E12F12G12H24A···24H24I24J24K24L
order1222233344444666668888888888121212121212121224···2424242424
size1166182241166182241212226666999922224412124···412121212

48 irreducible representations

dim111111111222222244444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6D6C4xS3C4xS3C8oD4S32C2xS32D12.C4C4xS32C24.D6
kernelC24.D6S3xC3:C8C12.31D6C3xC8:S3C8xC3:S3D6.D6D6:S3C3:D12C32:2Q8C8:S3C3:C8C24C4xS3Dic3D6C32C8C4C3C2C1
# reps121211242222244411424

Matrix representation of C24.D6 in GL6(F73)

2700000
0270000
0063000
0001000
00004627
0000460
,
7210000
7200000
000100
001000
0000721
000001
,
100000
1720000
0046000
0002700
0000721
000001

G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,63,0,0,0,0,0,0,10,0,0,0,0,0,0,46,46,0,0,0,0,27,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,1,1] >;

C24.D6 in GAP, Magma, Sage, TeX

C_{24}.D_6
% in TeX

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

G:=SmallGroup(288,453);
// by ID

G=gap.SmallGroup(288,453);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,422,219,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^12,b*a*b^-1=a^5,c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations

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