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G = D64  order 128 = 27

Dihedral group

p-group, metacyclic, nilpotent (class 6), monomial

Aliases: D64, C64⋊1C2, D32⋊1C2, C2.3D32, C16.5D4, C4.1D16, C8.10D8, C32.2C22, 2-Sylow(PGL(2,191)), sometimes denoted D128 or Dih64 or Dih128, SmallGroup(128,161)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C32 — D64
Chief series
C1 — C2 — C4 — C8 — C16 — C32 — D32 — D64
Lower central
C1 — C2 — C4 — C8 — C16 — C32 — D64
Upper central
C1 — C2 — C4 — C8 — C16 — C32 — D64
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C4 — C8 — C8 — C8 — C8 — C16 — C16 — C32 — D64

Generators and relations for D64
 G = < a,b | a64=b2=1, bab=a-1 >

C1
C2
32C2
32C2
C4
16C22
16C22
C8
8D4
8D4
C16
4D8
4D8
C32
2D16
2D16
D32
C64
D32
D64

Smallest permutation representation of D64
►On 64 points
Generators in S64
(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)

(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)
 
G:=sub<Sym(64)| (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), (2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)>;
 
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), (2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34) );
 
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)], [(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34)]])
 

35 conjugacy classes

class 1 2A2B2C 4 8A8B16A16B16C16D32A···32H64A···64P
order12224881616161632···3264···64
size11323222222222···22···2

35 irreducible representations

dim11122222
type++++++++
imageC1C2C2D4D8D16D32D64
kernelD64C64D32C16C8C4C2C1
# reps112124816

Matrix representation of D64 ►in GL2(𝔽193) generated by

4535
7910
,
10
192192
G:=sub<GL(2,GF(193))| [45,79,35,10],[1,192,0,192] >;
 

D64 in GAP, Magma, Sage, TeX 

D_{64}
 
% in TeX
 
G:=Group("D64");
 
// GroupNames label
 
G:=SmallGroup(128,161);
 
// by ID
 
G=gap.SmallGroup(128,161);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,85,254,135,142,675,346,192,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b|a^64=b^2=1,b*a*b=a^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D64 in TeX

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