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G = D96  order 192 = 26·3

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D96, C3⋊1D32, C96⋊1C2, C32⋊1S3, D48⋊1C2, C6.1D16, C8.5D12, C4.1D24, C2.3D48, C12.26D8, C16.13D6, C24.55D4, C48.14C22, sometimes denoted D192 or Dih96 or Dih192, SmallGroup(192,7)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C48 — D96
Chief series
C1 — C3 — C6 — C12 — C24 — C48 — D48 — D96
Lower central
C3 — C6 — C12 — C24 — C48 — D96
Upper central
C1 — C2 — C4 — C8 — C16 — C32

Generators and relations for D96
 G = < a,b | a96=b2=1, bab=a-1 >

C1
C2
48C2
48C2
C3
C4
24C22
24C22
C6
16S3
16S3
C8
12D4
12D4
C12
8D6
8D6
C16
6D8
6D8
C24
4D12
4D12
C32
3D16
3D16
C48
2D24
2D24
3D32
D48
C96
D48
D96

Smallest permutation representation of D96
►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 96)(2 95)(3 94)(4 93)(5 92)(6 91)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 81)(17 80)(18 79)(19 78)(20 77)(21 76)(22 75)(23 74)(24 73)(25 72)(26 71)(27 70)(28 69)(29 68)(30 67)(31 66)(32 65)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)
 
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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)>;
 
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), (1,96)(2,95)(3,94)(4,93)(5,92)(6,91)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,81)(17,80)(18,79)(19,78)(20,77)(21,76)(22,75)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49) );
 
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)], [(1,96),(2,95),(3,94),(4,93),(5,92),(6,91),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,81),(17,80),(18,79),(19,78),(20,77),(21,76),(22,75),(23,74),(24,73),(25,72),(26,71),(27,70),(28,69),(29,68),(30,67),(31,66),(32,65),(33,64),(34,63),(35,62),(36,61),(37,60),(38,59),(39,58),(40,57),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49)]])
 

51 conjugacy classes

class 1 2A2B2C 3  4  6 8A8B12A12B16A16B16C16D24A24B24C24D32A···32H48A···48H96A···96P
order1222346881212161616162424242432···3248···4896···96
size1148482222222222222222···22···22···2

51 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3D4D6D8D12D16D24D32D48D96
kernelD96C96D48C32C24C16C12C8C6C4C3C2C1
# reps11211122448816

Matrix representation of D96 ►in GL2(𝔽97) generated by

8931
6623
,
5033
8347
G:=sub<GL(2,GF(97))| [89,66,31,23],[50,83,33,47] >;
 

D96 in GAP, Magma, Sage, TeX 

D_{96}
 
% in TeX
 
G:=Group("D96");
 
// GroupNames label
 
G:=SmallGroup(192,7);
 
// by ID
 
G=gap.SmallGroup(192,7);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,85,92,254,142,675,192,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b|a^96=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D96 in TeX

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