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G = C24.3D6order 288 = 25·32

3rd non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C24.3D6, D6.6D12, D12.18D6, Dic6.15D6, Dic3.8D12, C8.1S32, C3⋊C8.1D6, C6.6(S3×D4), C24⋊C22S3, C8⋊S32S3, (C4×S3).3D6, (S3×C6).3D4, C6.6(C2×D12), (S3×Dic6)⋊2C2, C2.11(S3×D12), C325Q162C2, C31(C8.D6), C31(D4.D6), (C3×C24).4C22, (C3×Dic3).3D4, C323Q163C2, D125S3.1C2, D12.S32C2, (S3×C12).5C22, (C3×C12).48C23, (C3×D12).4C22, C323(C8.C22), C12.122(C22×S3), (C3×Dic6).2C22, C324Q8.2C22, C4.45(C2×S32), (C3×C8⋊S3)⋊4C2, (C3×C24⋊C2)⋊6C2, (C3×C6).32(C2×D4), (C3×C3⋊C8).1C22, SmallGroup(288,448)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C24.3D6
C1C3C32C3×C6C3×C12S3×C12S3×Dic6 — C24.3D6
C32C3×C6C3×C12 — C24.3D6
C1C2C4C8

Generators and relations for C24.3D6
 G = < a,b,c | a24=1, b6=c2=a12, bab-1=a11, cac-1=a-1, cbc-1=b5 >

Subgroups: 530 in 129 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×4], C22 [×2], S3 [×2], C6 [×2], C6 [×3], C8, C8, C2×C4 [×3], D4 [×2], Q8 [×4], C32, Dic3, Dic3 [×7], C12 [×2], C12 [×3], D6, D6, C2×C6 [×2], M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C3×S3 [×2], C3×C6, C3⋊C8, C24 [×2], C24 [×2], Dic6, Dic6 [×8], C4×S3, C4×S3 [×2], D12, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12, C3×D4, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3⋊Dic3 [×2], C3×C12, S3×C6, S3×C6, C8⋊S3, C24⋊C2, C24⋊C2, Dic12 [×4], D4.S3, C3⋊Q16, C3×M4(2), C3×SD16, C2×Dic6, C4○D12, D42S3, S3×Q8, C3×C3⋊C8, C3×C24, S3×Dic3 [×2], D6⋊S3, C322Q8, C3×Dic6, S3×C12, C3×D12, C324Q8 [×2], C8.D6, D4.D6, D12.S3, C323Q16, C3×C8⋊S3, C3×C24⋊C2, C325Q16, S3×Dic6, D125S3, C24.3D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, D12 [×2], C22×S3 [×2], C8.C22, S32, C2×D12, S3×D4, C2×S32, C8.D6, D4.D6, S3×D12, C24.3D6

Smallest permutation representation of C24.3D6
On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 44 5 40 9 36 13 32 17 28 21 48)(2 31 6 27 10 47 14 43 18 39 22 35)(3 42 7 38 11 34 15 30 19 26 23 46)(4 29 8 25 12 45 16 41 20 37 24 33)(49 83 69 87 65 91 61 95 57 75 53 79)(50 94 70 74 66 78 62 82 58 86 54 90)(51 81 71 85 67 89 63 93 59 73 55 77)(52 92 72 96 68 76 64 80 60 84 56 88)
(1 86 13 74)(2 85 14 73)(3 84 15 96)(4 83 16 95)(5 82 17 94)(6 81 18 93)(7 80 19 92)(8 79 20 91)(9 78 21 90)(10 77 22 89)(11 76 23 88)(12 75 24 87)(25 65 37 53)(26 64 38 52)(27 63 39 51)(28 62 40 50)(29 61 41 49)(30 60 42 72)(31 59 43 71)(32 58 44 70)(33 57 45 69)(34 56 46 68)(35 55 47 67)(36 54 48 66)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,44,5,40,9,36,13,32,17,28,21,48)(2,31,6,27,10,47,14,43,18,39,22,35)(3,42,7,38,11,34,15,30,19,26,23,46)(4,29,8,25,12,45,16,41,20,37,24,33)(49,83,69,87,65,91,61,95,57,75,53,79)(50,94,70,74,66,78,62,82,58,86,54,90)(51,81,71,85,67,89,63,93,59,73,55,77)(52,92,72,96,68,76,64,80,60,84,56,88), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,65,37,53)(26,64,38,52)(27,63,39,51)(28,62,40,50)(29,61,41,49)(30,60,42,72)(31,59,43,71)(32,58,44,70)(33,57,45,69)(34,56,46,68)(35,55,47,67)(36,54,48,66)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,44,5,40,9,36,13,32,17,28,21,48)(2,31,6,27,10,47,14,43,18,39,22,35)(3,42,7,38,11,34,15,30,19,26,23,46)(4,29,8,25,12,45,16,41,20,37,24,33)(49,83,69,87,65,91,61,95,57,75,53,79)(50,94,70,74,66,78,62,82,58,86,54,90)(51,81,71,85,67,89,63,93,59,73,55,77)(52,92,72,96,68,76,64,80,60,84,56,88), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,65,37,53)(26,64,38,52)(27,63,39,51)(28,62,40,50)(29,61,41,49)(30,60,42,72)(31,59,43,71)(32,58,44,70)(33,57,45,69)(34,56,46,68)(35,55,47,67)(36,54,48,66) );

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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,44,5,40,9,36,13,32,17,28,21,48),(2,31,6,27,10,47,14,43,18,39,22,35),(3,42,7,38,11,34,15,30,19,26,23,46),(4,29,8,25,12,45,16,41,20,37,24,33),(49,83,69,87,65,91,61,95,57,75,53,79),(50,94,70,74,66,78,62,82,58,86,54,90),(51,81,71,85,67,89,63,93,59,73,55,77),(52,92,72,96,68,76,64,80,60,84,56,88)], [(1,86,13,74),(2,85,14,73),(3,84,15,96),(4,83,16,95),(5,82,17,94),(6,81,18,93),(7,80,19,92),(8,79,20,91),(9,78,21,90),(10,77,22,89),(11,76,23,88),(12,75,24,87),(25,65,37,53),(26,64,38,52),(27,63,39,51),(28,62,40,50),(29,61,41,49),(30,60,42,72),(31,59,43,71),(32,58,44,70),(33,57,45,69),(34,56,46,68),(35,55,47,67),(36,54,48,66)])

36 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E6A6B6C6D6E8A8B12A12B12C12D12E12F12G24A···24H24I24J
order12223334444466666881212121212121224···242424
size116122242612363622412244122244412244···41212

36 irreducible representations

dim111111112222222222244444444
type+++++++++++++++++++-+++--+-
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6D6D6D12D12C8.C22S32S3×D4C2×S32C8.D6D4.D6S3×D12C24.3D6
kernelC24.3D6D12.S3C323Q16C3×C8⋊S3C3×C24⋊C2C325Q16S3×Dic6D125S3C8⋊S3C24⋊C2C3×Dic3S3×C6C3⋊C8C24Dic6C4×S3D12Dic3D6C32C8C6C4C3C3C2C1
# reps111111111111121112211112224

Matrix representation of C24.3D6 in GL4(𝔽73) generated by

235500
18500
17385055
32311868
,
132821
132812
6665550
6675550
,
121000
226100
26501061
42625163
G:=sub<GL(4,GF(73))| [23,18,17,32,55,5,38,31,0,0,50,18,0,0,55,68],[13,13,66,66,28,28,6,7,2,1,55,55,1,2,50,50],[12,22,26,42,10,61,50,62,0,0,10,51,0,0,61,63] >;

C24.3D6 in GAP, Magma, Sage, TeX

C_{24}._3D_6
% in TeX

G:=Group("C24.3D6");
// GroupNames label

G:=SmallGroup(288,448);
// by ID

G=gap.SmallGroup(288,448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,135,58,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=1,b^6=c^2=a^12,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=b^5>;
// generators/relations

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