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G = D96order 192 = 26·3

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D96, C31D32, C961C2, C321S3, D481C2, 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

C1C48 — D96
C1C3C6C12C24C48D48 — D96
C3C6C12C24C48 — D96
C1C2C4C8C16C32

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

48C2
48C2
24C22
24C22
16S3
16S3
12D4
12D4
8D6
8D6
6D8
6D8
4D12
4D12
3D16
3D16
2D24
2D24
3D32

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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