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G = C2xSD64  order 128 = 27

Direct product of C2 and SD64

direct product, p-group, metabelian, nilpotent (class 5), monomial

Aliases: C2xSD64, C4.7D16, C8.19D8, C32:3C22, C16.10D4, C16.7C23, Q32:1C22, D16.1C22, C22.15D16, (C2xC32):7C2, (C2xQ32):7C2, C4.14(C2xD8), C8.46(C2xD4), (C2xC4).89D8, (C2xD16).4C2, C2.13(C2xD16), (C2xC8).258D4, (C2xC16).89C22, 2-Sylow(GU(3,17)), SmallGroup(128,992)

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

Derived series
C1 — C16 — C2xSD64
Chief series
C1 — C2 — C4 — C8 — C16 — C2xC16 — C2xD16 — C2xSD64
Lower central
C1 — C2 — C4 — C8 — C16 — C2xSD64
Upper central
C1 — C22 — C2xC4 — C2xC8 — C2xC16 — C2xSD64
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — C2xSD64

Generators and relations for C2xSD64
 G = < a,b,c | a2=b32=c2=1, ab=ba, ac=ca, cbc=b15 >

Subgroups: 204 in 56 conjugacy classes, 28 normal (16 characteristic)
Quotients: C1, C2, C22, D4, C23, D8, C2xD4, D16, C2xD8, SD64, C2xD16, C2xSD64
C1
C2
C2
C2
16C2
16C2
C4
C22
C4
8C4
8C22
8C4
8C22
16C22
16C22
C8
C8
C2xC4
4Q8
4D4
4D4
4Q8
8D4
8C2xC4
8Q8
8C23
C2xC8
C16
C16
2Q16
2D8
2Q16
2D8
4D8
4Q16
4C2xD4
4C2xQ8
C32
C2xC16
Q32
C32
D16
Q32
D16
2C2xQ16
2Q32
2C2xD8
2D16
C2xQ32
C2xC32
C2xD16
SD64
SD64
SD64
SD64
C2xSD64

Smallest permutation representation of C2xSD64
►On 64 points
Generators in S64
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 44)(26 45)(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)

(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)

(1 36)(2 51)(3 34)(4 49)(5 64)(6 47)(7 62)(8 45)(9 60)(10 43)(11 58)(12 41)(13 56)(14 39)(15 54)(16 37)(17 52)(18 35)(19 50)(20 33)(21 48)(22 63)(23 46)(24 61)(25 44)(26 59)(27 42)(28 57)(29 40)(30 55)(31 38)(32 53)
 
G:=sub<Sym(64)| (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51), (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), (1,36)(2,51)(3,34)(4,49)(5,64)(6,47)(7,62)(8,45)(9,60)(10,43)(11,58)(12,41)(13,56)(14,39)(15,54)(16,37)(17,52)(18,35)(19,50)(20,33)(21,48)(22,63)(23,46)(24,61)(25,44)(26,59)(27,42)(28,57)(29,40)(30,55)(31,38)(32,53)>;
 
G:=Group( (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(30,49)(31,50)(32,51), (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), (1,36)(2,51)(3,34)(4,49)(5,64)(6,47)(7,62)(8,45)(9,60)(10,43)(11,58)(12,41)(13,56)(14,39)(15,54)(16,37)(17,52)(18,35)(19,50)(20,33)(21,48)(22,63)(23,46)(24,61)(25,44)(26,59)(27,42)(28,57)(29,40)(30,55)(31,38)(32,53) );
 
G=PermutationGroup([[(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(21,40),(22,41),(23,42),(24,43),(25,44),(26,45),(27,46),(28,47),(29,48),(30,49),(31,50),(32,51)], [(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)], [(1,36),(2,51),(3,34),(4,49),(5,64),(6,47),(7,62),(8,45),(9,60),(10,43),(11,58),(12,41),(13,56),(14,39),(15,54),(16,37),(17,52),(18,35),(19,50),(20,33),(21,48),(22,63),(23,46),(24,61),(25,44),(26,59),(27,42),(28,57),(29,40),(30,55),(31,38),(32,53)]])
 

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D8A8B8C8D16A···16H32A···32P
order1222224444888816···1632···32
size1111161622161622222···22···2

38 irreducible representations

dim111112222222
type+++++++++++
imageC1C2C2C2C2D4D4D8D8D16D16SD64
kernelC2xSD64C2xC32SD64C2xD16C2xQ32C16C2xC8C8C2xC4C4C22C2
# reps1141111224416

Matrix representation of C2xSD64 ►in GL3(F97) generated by

9600
0960
0096
,
100
03635
06236
,
100
010
0096
G:=sub<GL(3,GF(97))| [96,0,0,0,96,0,0,0,96],[1,0,0,0,36,62,0,35,36],[1,0,0,0,1,0,0,0,96] >;
 

C2xSD64 in GAP, Magma, Sage, TeX 

C_2\times {\rm SD}_{64}
 
% in TeX
 
G:=Group("C2xSD64");
 
// GroupNames label
 
G:=SmallGroup(128,992);
 
// by ID
 
G=gap.SmallGroup(128,992);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,-2,-2,-2,448,141,675,346,192,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^2=b^32=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^15>;
 
// generators/relations
 

Export

Subgroup lattice of C2xSD64 in TeX

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