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G = C6.16D24order 288 = 25·32

5th non-split extension by C6 of D24 acting via D24/D12=C2

metabelian, supersoluble, monomial

Aliases: C6.16D24, D121Dic3, C62.18D4, (C3×D12)⋊2C4, (C3×C6).14D8, C12.14(C4×S3), C6.5(D4⋊S3), (C2×D12).1S3, (C6×D12).3C2, (C2×C12).76D6, (C3×C12).33D4, (C2×C6).49D12, (C3×C6).8SD16, C4.1(S3×Dic3), C6.35(D6⋊C4), C33(C2.D24), C6.1(D4.S3), C6.10(C24⋊C2), C31(D4⋊Dic3), C2.1(C3⋊D24), C12.29(C3⋊D4), C325(D4⋊C4), C2.6(D6⋊Dic3), (C6×C12).25C22, C12.22(C2×Dic3), C12⋊Dic313C2, C4.11(D6⋊S3), C6.5(C6.D4), C2.1(D12.S3), C22.12(C3⋊D12), (C6×C3⋊C8)⋊3C2, (C2×C3⋊C8)⋊1S3, (C2×C4).52S32, (C3×C12).23(C2×C4), (C2×C6).30(C3⋊D4), (C3×C6).30(C22⋊C4), SmallGroup(288,211)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C6.16D24
C1C3C32C3×C6C3×C12C6×C12C6×D12 — C6.16D24
C32C3×C6C3×C12 — C6.16D24
C1C22C2×C4

Generators and relations for C6.16D24
 G = < a,b,c | a6=b24=1, c2=a3, bab-1=cac-1=a-1, cbc-1=a3b-1 >

Subgroups: 458 in 113 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×2], C4, C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C8, C2×C4, C2×C4, D4 [×3], C23, C32, Dic3 [×4], C12 [×4], C12 [×2], D6 [×4], C2×C6 [×2], C2×C6 [×5], C4⋊C4, C2×C8, C2×D4, C3×S3 [×2], C3×C6 [×3], C3⋊C8, C24, D12 [×2], D12, C2×Dic3 [×4], C2×C12 [×2], C2×C12, C3×D4 [×3], C22×S3, C22×C6, D4⋊C4, C3⋊Dic3, C3×C12 [×2], S3×C6 [×4], C62, C2×C3⋊C8, C4⋊Dic3 [×3], C2×C24, C2×D12, C6×D4, C3×C3⋊C8, C3×D12 [×2], C3×D12, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C2.D24, D4⋊Dic3, C6×C3⋊C8, C12⋊Dic3, C6×D12, C6.16D24
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], Dic3 [×2], D6 [×2], C22⋊C4, D8, SD16, C4×S3, D12, C2×Dic3, C3⋊D4 [×3], D4⋊C4, S32, C24⋊C2, D24, D6⋊C4, D4⋊S3, D4.S3, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C2.D24, D4⋊Dic3, C3⋊D24, D12.S3, D6⋊Dic3, C6.16D24

Smallest permutation representation of C6.16D24
On 96 points
Generators in S96
(1 66 9 50 17 58)(2 59 18 51 10 67)(3 68 11 52 19 60)(4 61 20 53 12 69)(5 70 13 54 21 62)(6 63 22 55 14 71)(7 72 15 56 23 64)(8 65 24 57 16 49)(25 73 33 81 41 89)(26 90 42 82 34 74)(27 75 35 83 43 91)(28 92 44 84 36 76)(29 77 37 85 45 93)(30 94 46 86 38 78)(31 79 39 87 47 95)(32 96 48 88 40 80)
(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 95 50 39)(2 38 51 94)(3 93 52 37)(4 36 53 92)(5 91 54 35)(6 34 55 90)(7 89 56 33)(8 32 57 88)(9 87 58 31)(10 30 59 86)(11 85 60 29)(12 28 61 84)(13 83 62 27)(14 26 63 82)(15 81 64 25)(16 48 65 80)(17 79 66 47)(18 46 67 78)(19 77 68 45)(20 44 69 76)(21 75 70 43)(22 42 71 74)(23 73 72 41)(24 40 49 96)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A···6F6G6H6I6J6K6L6M8A8B8C8D12A12B12C12D12E···12J24A···24H
order12222233344446···6666666688881212121212···1224···24
size111112122242236362···244412121212666622224···46···6

48 irreducible representations

dim111112222222222222244444444
type++++++++-++++++---++-
imageC1C2C2C2C4S3S3D4D4Dic3D6D8SD16C4×S3C3⋊D4D12C3⋊D4C24⋊C2D24S32D4⋊S3D4.S3S3×Dic3D6⋊S3C3⋊D12C3⋊D24D12.S3
kernelC6.16D24C6×C3⋊C8C12⋊Dic3C6×D12C3×D12C2×C3⋊C8C2×D12C3×C12C62D12C2×C12C3×C6C3×C6C12C12C2×C6C2×C6C6C6C2×C4C6C6C4C4C22C2C2
# reps111141111222224224411111122

Matrix representation of C6.16D24 in GL6(𝔽73)

100000
010000
0072100
0072000
0000720
0000072
,
10350000
0220000
000100
001000
00006043
00003030
,
66200000
5670000
0007200
0072000
00004627
0000027

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[10,0,0,0,0,0,35,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,30,0,0,0,0,43,30],[66,56,0,0,0,0,20,7,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,27,27] >;

C6.16D24 in GAP, Magma, Sage, TeX

C_6._{16}D_{24}
% in TeX

G:=Group("C6.16D24");
// GroupNames label

G:=SmallGroup(288,211);
// by ID

G=gap.SmallGroup(288,211);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,85,92,422,100,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^6=b^24=1,c^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^3*b^-1>;
// generators/relations

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