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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⋊S36S3, D6.1(C4×S3), (C4×S3).30D6, C323(C8○D4), C32(D12.C4), C3⋊D12.2C4, D6⋊S3.2C4, Dic3.2(C4×S3), C322Q8.2C4, (S3×C12).7C22, (C3×C24).47C22, D6.D6.3C2, C12.31D610C2, C12.139(C22×S3), (C3×C12).140C23, C324C8.37C22, C2.8(C4×S32), C6.6(S3×C2×C4), (S3×C3⋊C8)⋊11C2, C4.86(C2×S32), (C8×C3⋊S3)⋊14C2, (S3×C6).3(C2×C4), (C3×C8⋊S3)⋊12C2, (C3×C3⋊C8).25C22, (C3×C6).6(C22×C4), (C4×C3⋊S3).87C22, C3⋊Dic3.32(C2×C4), (C3×Dic3).3(C2×C4), (C2×C3⋊S3).28(C2×C4), SmallGroup(288,453)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.D6
C1C3C32C3×C6C3×C12S3×C12D6.D6 — C24.D6
C32C3×C6 — 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 [×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 [×5], C8⋊S3 [×2], C8⋊S3 [×2], C2×C3⋊C8 [×2], C3×M4(2) [×2], C4○D12 [×2], C3×C3⋊C8 [×2], C324C8, C3×C24, D6⋊S3, C3⋊D12 [×2], C322Q8, S3×C12 [×2], C4×C3⋊S3, D12.C4 [×2], S3×C3⋊C8 [×2], C12.31D6, C3×C8⋊S3 [×2], C8×C3⋊S3, D6.D6, C24.D6
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, D12.C4 [×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 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++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6D6C4×S3C4×S3C8○D4S32C2×S32D12.C4C4×S32C24.D6
kernelC24.D6S3×C3⋊C8C12.31D6C3×C8⋊S3C8×C3⋊S3D6.D6D6⋊S3C3⋊D12C322Q8C8⋊S3C3⋊C8C24C4×S3Dic3D6C32C8C4C3C2C1
# 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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