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G = C96⋊C2order 192 = 26·3

6th semidirect product of C96 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C966C2, C323S3, D6.C16, C31M6(2), C16.20D6, Dic3.C16, C48.25C22, C3⋊C324C2, C3⋊C8.2C8, C3⋊C16.2C4, (S3×C8).2C4, (C4×S3).2C8, C4.17(S3×C8), C8.37(C4×S3), C2.3(S3×C16), C6.2(C2×C16), (S3×C16).2C2, C24.58(C2×C4), C12.22(C2×C8), SmallGroup(192,6)

Series: Derived Chief Lower central Upper central

C1C6 — C96⋊C2
C1C3C6C12C24C48S3×C16 — C96⋊C2
C3C6 — C96⋊C2
C1C16C32

Generators and relations for C96⋊C2
 G = < a,b | a96=b2=1, bab=a17 >

6C2
3C22
3C4
2S3
3C2×C4
3C8
3C2×C8
3C16
3C2×C16
3C32
3M6(2)

Smallest permutation representation of C96⋊C2
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)
(2 18)(3 35)(4 52)(5 69)(6 86)(8 24)(9 41)(10 58)(11 75)(12 92)(14 30)(15 47)(16 64)(17 81)(20 36)(21 53)(22 70)(23 87)(26 42)(27 59)(28 76)(29 93)(32 48)(33 65)(34 82)(38 54)(39 71)(40 88)(44 60)(45 77)(46 94)(50 66)(51 83)(56 72)(57 89)(62 78)(63 95)(68 84)(74 90)(80 96)

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

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

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

72 conjugacy classes

class 1 2A2B 3 4A4B4C 6 8A8B8C8D8E8F12A12B16A···16H16I16J16K16L24A24B24C24D32A···32H32I···32P48A···48H96A···96P
order12234446888888121216···16161616162424242432···3232···3248···4896···96
size11621162111166221···1666622222···26···62···22···2

72 irreducible representations

dim11111111112222222
type++++++
imageC1C2C2C2C4C4C8C8C16C16S3D6C4×S3S3×C8M6(2)S3×C16C96⋊C2
kernelC96⋊C2C3⋊C32C96S3×C16C3⋊C16S3×C8C3⋊C8C4×S3Dic3D6C32C16C8C4C3C2C1
# reps111122448811248816

Matrix representation of C96⋊C2 in GL2(𝔽17) generated by

1510
10
,
12
016
G:=sub<GL(2,GF(17))| [15,1,10,0],[1,0,2,16] >;

C96⋊C2 in GAP, Magma, Sage, TeX

C_{96}\rtimes C_2
% in TeX

G:=Group("C96:C2");
// GroupNames label

G:=SmallGroup(192,6);
// by ID

G=gap.SmallGroup(192,6);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,36,58,80,102,6278]);
// Polycyclic

G:=Group<a,b|a^96=b^2=1,b*a*b=a^17>;
// generators/relations

Export

Subgroup lattice of C96⋊C2 in TeX

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