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G = C24⋊D6order 288 = 25·32

14th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C2414D6, C89S32, C3⋊C818D6, C8⋊S35S3, D6.5(C4×S3), (C4×S3).28D6, C3⋊S31M4(2), C32(S3×M4(2)), (C3×C24)⋊21C22, Dic3.8(C4×S3), (S3×Dic3).1C4, C6.D6.3C4, C322(C2×M4(2)), C12.29D68C2, D6.Dic311C2, (S3×C12).2C22, C12.136(C22×S3), (C3×C12).137C23, C324C826C22, C2.5(C4×S32), C6.3(S3×C2×C4), (C2×S32).4C4, (C4×S32).2C2, C4.83(C2×S32), (C8×C3⋊S3)⋊13C2, (C3×C3⋊C8)⋊23C22, (S3×C6).1(C2×C4), (C3×C8⋊S3)⋊10C2, (C3×C6).3(C22×C4), (C4×C3⋊S3).85C22, C3⋊Dic3.30(C2×C4), (C3×Dic3).2(C2×C4), (C2×C3⋊S3).26(C2×C4), SmallGroup(288,439)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24⋊D6
C1C3C32C3×C6C3×C12S3×C12C4×S32 — C24⋊D6
C32C3×C6 — C24⋊D6
C1C4C8

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

Subgroups: 498 in 147 conjugacy classes, 52 normal (18 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4, C4 [×3], C22 [×5], S3 [×8], C6 [×2], C6 [×3], C8, C8 [×3], C2×C4 [×6], C23, C32, Dic3 [×2], Dic3 [×3], C12 [×2], C12 [×3], D6 [×2], D6 [×7], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3×S3 [×2], C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C3⋊C8 [×3], C24 [×2], C24 [×3], C4×S3 [×2], C4×S3 [×7], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×M4(2), C3×Dic3 [×2], C3⋊Dic3, C3×C12, S32 [×2], S3×C6 [×2], C2×C3⋊S3, S3×C8 [×5], C8⋊S3 [×2], C8⋊S3 [×2], C4.Dic3 [×2], C3×M4(2) [×2], S3×C2×C4 [×2], C3×C3⋊C8 [×2], C324C8, C3×C24, S3×Dic3 [×2], C6.D6, S3×C12 [×2], C4×C3⋊S3, C2×S32, S3×M4(2) [×2], C12.29D6, D6.Dic3 [×2], C3×C8⋊S3 [×2], C8×C3⋊S3, C4×S32, C24⋊D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], M4(2) [×2], C22×C4, C4×S3 [×4], C22×S3 [×2], C2×M4(2), S32, S3×C2×C4 [×2], C2×S32, S3×M4(2) [×2], C4×S32, 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 17)(2 10)(4 20)(5 13)(7 23)(8 16)(11 19)(14 22)(25 33)(27 43)(28 36)(30 46)(31 39)(34 42)(37 45)(40 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), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48)>;

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), (1,17)(2,10)(4,20)(5,13)(7,23)(8,16)(11,19)(14,22)(25,33)(27,43)(28,36)(30,46)(31,39)(34,42)(37,45)(40,48) );

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)], [(1,17),(2,10),(4,20),(5,13),(7,23),(8,16),(11,19),(14,22),(25,33),(27,43),(28,36),(30,46),(31,39),(34,42),(37,45),(40,48)])

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H24A···24H24I24J24K24L
order1222223334444446666688888888121212121212121224···2424242424
size1166992241166992241212226666181822224412124···412121212

48 irreducible representations

dim111111111222222244444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6D6M4(2)C4×S3C4×S3S32C2×S32S3×M4(2)C4×S32C24⋊D6
kernelC24⋊D6C12.29D6D6.Dic3C3×C8⋊S3C8×C3⋊S3C4×S32S3×Dic3C6.D6C2×S32C8⋊S3C3⋊C8C24C4×S3C3⋊S3Dic3D6C8C4C3C2C1
# reps112211422222244411424

Matrix representation of C24⋊D6 in GL6(𝔽73)

100000
010000
0051300
0002200
0000721
0000720
,
7210000
7200000
0058500
00721500
000001
000010
,
7200000
7210000
001000
000100
000001
000010

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,51,0,0,0,0,0,3,22,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,72,0,0,0,0,1,0,0,0,0,0,0,0,58,72,0,0,0,0,5,15,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,72,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24⋊D6 in GAP, Magma, Sage, TeX

C_{24}\rtimes D_6
% in TeX

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

G:=SmallGroup(288,439);
// by ID

G=gap.SmallGroup(288,439);
# by ID

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

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

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