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## G = SD128order 128 = 27

### Semidihedral group

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

Aliases: SD128, C642C2, D32.C2, Q641C2, C4.2D16, C16.6D4, C8.11D8, C2.4D32, C32.3C22, 2-Sylow(GL(2,31)), also known as the quasi-dihedral group QD128, SmallGroup(128,162)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C32 — SD128
 Chief series C1 — C2 — C4 — C8 — C16 — C32 — D32 — SD128
 Lower central C1 — C2 — C4 — C8 — C16 — C32 — SD128
 Upper central C1 — C2 — C4 — C8 — C16 — C32 — SD128
 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 — SD128

Generators and relations for SD128
G = < a,b | a64=b2=1, bab=a31 >

Smallest permutation representation of SD128
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 32)(3 63)(4 30)(5 61)(6 28)(7 59)(8 26)(9 57)(10 24)(11 55)(12 22)(13 53)(14 20)(15 51)(16 18)(17 49)(19 47)(21 45)(23 43)(25 41)(27 39)(29 37)(31 35)(34 64)(36 62)(38 60)(40 58)(42 56)(44 54)(46 52)(48 50)

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

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

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

35 conjugacy classes

 class 1 2A 2B 4A 4B 8A 8B 16A 16B 16C 16D 32A ··· 32H 64A ··· 64P order 1 2 2 4 4 8 8 16 16 16 16 32 ··· 32 64 ··· 64 size 1 1 32 2 32 2 2 2 2 2 2 2 ··· 2 2 ··· 2

35 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 D4 D8 D16 D32 SD128 kernel SD128 C64 D32 Q64 C16 C8 C4 C2 C1 # reps 1 1 1 1 1 2 4 8 16

Matrix representation of SD128 in GL2(𝔽31) generated by

 12 30 30 0
,
 1 12 0 30
G:=sub<GL(2,GF(31))| [12,30,30,0],[1,0,12,30] >;

SD128 in GAP, Magma, Sage, TeX

{\rm SD}_{128}
% in TeX

G:=Group("SD128");
// GroupNames label

G:=SmallGroup(128,162);
// by ID

G=gap.SmallGroup(128,162);
# by ID

G:=PCGroup([7,-2,2,-2,-2,-2,-2,-2,448,85,254,135,142,675,346,192,1684,851,242,4037,2028,124]);
// Polycyclic

G:=Group<a,b|a^64=b^2=1,b*a*b=a^31>;
// generators/relations

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