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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, D12:1Dic3, C62.18D4, (C3xD12):2C4, (C3xC6).14D8, C12.14(C4xS3), C6.5(D4:S3), (C2xD12).1S3, (C6xD12).3C2, (C2xC12).76D6, (C3xC12).33D4, (C2xC6).49D12, (C3xC6).8SD16, C4.1(S3xDic3), C6.35(D6:C4), C3:3(C2.D24), C6.1(D4.S3), C6.10(C24:C2), C3:1(D4:Dic3), C2.1(C3:D24), C12.29(C3:D4), C32:5(D4:C4), C2.6(D6:Dic3), (C6xC12).25C22, C12.22(C2xDic3), C12:Dic3:13C2, C4.11(D6:S3), C6.5(C6.D4), C2.1(D12.S3), C22.12(C3:D12), (C6xC3:C8):3C2, (C2xC3:C8):1S3, (C2xC4).52S32, (C3xC12).23(C2xC4), (C2xC6).30(C3:D4), (C3xC6).30(C22:C4), SmallGroup(288,211)

Series: Derived Chief Lower central Upper central

C1C3xC12 — C6.16D24
C1C3C32C3xC6C3xC12C6xC12C6xD12 — C6.16D24
C32C3xC6C3xC12 — C6.16D24
C1C22C2xC4

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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C4:C4, C2xC8, C2xD4, C3xS3, C3xC6, C3:C8, C24, D12, D12, C2xDic3, C2xC12, C2xC12, C3xD4, C22xS3, C22xC6, D4:C4, C3:Dic3, C3xC12, S3xC6, C62, C2xC3:C8, C4:Dic3, C2xC24, C2xD12, C6xD4, C3xC3:C8, C3xD12, C3xD12, C2xC3:Dic3, C6xC12, S3xC2xC6, C2.D24, D4:Dic3, C6xC3:C8, C12:Dic3, C6xD12, C6.16D24
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, D8, SD16, C4xS3, D12, C2xDic3, C3:D4, D4:C4, S32, C24:C2, D24, D6:C4, D4:S3, D4.S3, C6.D4, S3xDic3, 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 71 9 55 17 63)(2 64 18 56 10 72)(3 49 11 57 19 65)(4 66 20 58 12 50)(5 51 13 59 21 67)(6 68 22 60 14 52)(7 53 15 61 23 69)(8 70 24 62 16 54)(25 90 41 82 33 74)(26 75 34 83 42 91)(27 92 43 84 35 76)(28 77 36 85 44 93)(29 94 45 86 37 78)(30 79 38 87 46 95)(31 96 47 88 39 80)(32 81 40 89 48 73)
(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 55 77)(2 76 56 43)(3 42 57 75)(4 74 58 41)(5 40 59 73)(6 96 60 39)(7 38 61 95)(8 94 62 37)(9 36 63 93)(10 92 64 35)(11 34 65 91)(12 90 66 33)(13 32 67 89)(14 88 68 31)(15 30 69 87)(16 86 70 29)(17 28 71 85)(18 84 72 27)(19 26 49 83)(20 82 50 25)(21 48 51 81)(22 80 52 47)(23 46 53 79)(24 78 54 45)

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim111112222222222222244444444
type++++++++-++++++---++-
imageC1C2C2C2C4S3S3D4D4Dic3D6D8SD16C4xS3C3:D4D12C3:D4C24:C2D24S32D4:S3D4.S3S3xDic3D6:S3C3:D12C3:D24D12.S3
kernelC6.16D24C6xC3:C8C12:Dic3C6xD12C3xD12C2xC3:C8C2xD12C3xC12C62D12C2xC12C3xC6C3xC6C12C12C2xC6C2xC6C6C6C2xC4C6C6C4C4C22C2C2
# 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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