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G = C24.63D6order 288 = 25·32

16th non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C24.63D6, C8.24S32, (S3×C8)⋊6S3, C3⋊C8.29D6, D6.2(C4×S3), (S3×C24)⋊12C2, C31(C8○D12), (C4×S3).40D6, C321(C8○D4), D6⋊S3.3C4, C3⋊D12.3C4, Dic3.5(C4×S3), C322Q8.3C4, D6.Dic315C2, (C3×C24).45C22, D6.D6.5C2, C12.31D613C2, (S3×C12).38C22, (C3×C12).138C23, C12.137(C22×S3), C324C8.36C22, C2.6(C4×S32), C6.4(S3×C2×C4), C4.84(C2×S32), (C8×C3⋊S3)⋊11C2, (S3×C6).10(C2×C4), (C3×C3⋊C8).37C22, (C3×C6).4(C22×C4), (C4×C3⋊S3).86C22, C3⋊Dic3.31(C2×C4), (C3×Dic3).11(C2×C4), (C2×C3⋊S3).27(C2×C4), SmallGroup(288,451)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.63D6
C1C3C32C3×C6C3×C12S3×C12D6.D6 — C24.63D6
C32C3×C6 — C24.63D6
C1C8

Generators and relations for C24.63D6
 G = < a,b,c | a24=b6=1, c2=a12, bab-1=cac-1=a17, cbc-1=a12b-1 >

Subgroups: 434 in 135 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×3], S3 [×6], C6 [×2], C6 [×3], C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×3], C2×C6 [×2], C2×C8 [×3], M4(2) [×3], C4○D4, C3×S3 [×2], C3⋊S3, C3×C6, C3⋊C8 [×2], C3⋊C8 [×3], C24 [×2], C24 [×3], Dic6 [×2], C4×S3 [×2], C4×S3 [×3], D12 [×2], C3⋊D4 [×4], C2×C12 [×2], C8○D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], C2×C3⋊S3, S3×C8 [×2], S3×C8 [×3], C8⋊S3 [×4], C4.Dic3 [×2], C2×C24 [×2], C4○D12 [×2], C3×C3⋊C8 [×2], C324C8, C3×C24, D6⋊S3, C3⋊D12 [×2], C322Q8, S3×C12 [×2], C4×C3⋊S3, C8○D12 [×2], D6.Dic3 [×2], C12.31D6, S3×C24 [×2], C8×C3⋊S3, D6.D6, C24.63D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4×S3 [×4], C22×S3 [×2], C8○D4, S32, S3×C2×C4 [×2], C2×S32, C8○D12 [×2], C4×S32, C24.63D6

Smallest permutation representation of C24.63D6
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 38 17 46 9 30)(2 31 18 39 10 47)(3 48 19 32 11 40)(4 41 20 25 12 33)(5 34 21 42 13 26)(6 27 22 35 14 43)(7 44 23 28 15 36)(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), (1,38,17,46,9,30)(2,31,18,39,10,47)(3,48,19,32,11,40)(4,41,20,25,12,33)(5,34,21,42,13,26)(6,27,22,35,14,43)(7,44,23,28,15,36)(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), (1,38,17,46,9,30)(2,31,18,39,10,47)(3,48,19,32,11,40)(4,41,20,25,12,33)(5,34,21,42,13,26)(6,27,22,35,14,43)(7,44,23,28,15,36)(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)], [(1,38,17,46,9,30),(2,31,18,39,10,47),(3,48,19,32,11,40),(4,41,20,25,12,33),(5,34,21,42,13,26),(6,27,22,35,14,43),(7,44,23,28,15,36),(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)])

60 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D6E6F6G8A8B8C8D8E8F8G8H8I8J12A12B12C12D12E12F12G12H12I12J24A···24H24I24J24K24L24M···24T
order1222233344444666666688888888881212121212121212121224···242424242424···24
size116618224116618224666611116666181822224466662···244446···6

60 irreducible representations

dim111111111222222224444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6D6C4×S3C4×S3C8○D4C8○D12S32C2×S32C4×S32C24.63D6
kernelC24.63D6D6.Dic3C12.31D6S3×C24C8×C3⋊S3D6.D6D6⋊S3C3⋊D12C322Q8S3×C8C3⋊C8C24C4×S3Dic3D6C32C3C8C4C2C1
# reps1212112422222444161124

Matrix representation of C24.63D6 in GL6(𝔽73)

100000
010000
0010000
0001000
0000027
00004646
,
1720000
100000
0014800
0007200
000001
000010
,
7200000
7210000
0046000
00652700
0000072
0000720

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,0,46,0,0,0,0,27,46],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,46,65,0,0,0,0,0,27,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

C24.63D6 in GAP, Magma, Sage, TeX

C_{24}._{63}D_6
% in TeX

G:=Group("C24.63D6");
// GroupNames label

G:=SmallGroup(288,451);
// by ID

G=gap.SmallGroup(288,451);
# by ID

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

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

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