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G = C24.61D6order 288 = 25·32

14th non-split extension by C24 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C24.61D6, C324M5(2), C3⋊C164S3, C8.22S32, D6.(C3⋊C8), (S3×C8).2S3, (S3×C6).2C8, C6.18(S3×C8), Dic3.(C3⋊C8), C3⋊C8.2Dic3, (S3×C24).4C2, (S3×C12).1C4, C12.95(C4×S3), C33(D6.C8), C24.S37C2, (C4×S3).2Dic3, (C3×Dic3).1C8, C4.17(S3×Dic3), C31(C12.C8), (C3×C24).43C22, C12.28(C2×Dic3), C6.2(C2×C3⋊C8), C2.3(S3×C3⋊C8), (C3×C3⋊C16)⋊9C2, (C3×C3⋊C8).4C4, (C3×C6).14(C2×C8), (C3×C12).79(C2×C4), SmallGroup(288,191)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C24.61D6
C1C3C32C3×C6C3×C12C3×C24S3×C24 — C24.61D6
C32C3×C6 — C24.61D6
C1C8

Generators and relations for C24.61D6
 G = < a,b,c | a24=b6=1, c2=a21, bab-1=cac-1=a17, cbc-1=a12b-1 >

Subgroups: 126 in 57 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, C8, C2×C4, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16 [×2], C2×C8, C3×S3, C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, M5(2), C3×Dic3, C3×C12, S3×C6, C3⋊C16, C3⋊C16 [×3], C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, D6.C8, C12.C8, C3×C3⋊C16, C24.S3, S3×C24, C24.61D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, Dic3 [×2], D6 [×2], C2×C8, C3⋊C8 [×2], C4×S3, C2×Dic3, M5(2), S32, S3×C8, C2×C3⋊C8, S3×Dic3, D6.C8, C12.C8, S3×C3⋊C8, C24.61D6

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

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

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

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

60 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C6D6E8A8B8C8D8E8F12A12B12C12D12E12F12G12H16A16B16C16D16E16F16G16H24A···24H24I24J24K24L24M24N24O24P48A···48H
order122333444666668888881212121212121212161616161616161624···24242424242424242448···48
size11622411622466111166222244666666181818182···2444466666···6

60 irreducible representations

dim111111112222222222224444
type++++++-+-+-
imageC1C2C2C2C4C4C8C8S3S3Dic3D6Dic3C3⋊C8C4×S3C3⋊C8M5(2)S3×C8D6.C8C12.C8S32S3×Dic3S3×C3⋊C8C24.61D6
kernelC24.61D6C3×C3⋊C16C24.S3S3×C24C3×C3⋊C8S3×C12C3×Dic3S3×C6C3⋊C16S3×C8C3⋊C8C24C4×S3Dic3C12D6C32C6C3C3C8C4C2C1
# reps111122441112122244881124

Matrix representation of C24.61D6 in GL6(𝔽97)

4700000
0470000
0002200
00757500
000010
000001
,
95800000
2320000
0009600
0096000
000001
00009696
,
74950000
94230000
0003300
0033000
000001
000010

G:=sub<GL(6,GF(97))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,0,75,0,0,0,0,22,75,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[95,23,0,0,0,0,80,2,0,0,0,0,0,0,0,96,0,0,0,0,96,0,0,0,0,0,0,0,0,96,0,0,0,0,1,96],[74,94,0,0,0,0,95,23,0,0,0,0,0,0,0,33,0,0,0,0,33,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C24.61D6 in GAP, Magma, Sage, TeX

C_{24}._{61}D_6
% in TeX

G:=Group("C24.61D6");
// GroupNames label

G:=SmallGroup(288,191);
// by ID

G=gap.SmallGroup(288,191);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,36,58,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^24=b^6=1,c^2=a^21,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations

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