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G = C2×SD64order 128 = 27

Direct product of C2 and SD64

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

Aliases: C2×SD64, C4.7D16, C8.19D8, C323C22, C16.10D4, C16.7C23, Q321C22, D16.1C22, C22.15D16, (C2×C32)⋊7C2, (C2×Q32)⋊7C2, C4.14(C2×D8), C8.46(C2×D4), (C2×C4).89D8, (C2×D16).4C2, C2.13(C2×D16), (C2×C8).258D4, (C2×C16).89C22, 2-Sylow(GU(3,17)), SmallGroup(128,992)

Series: Derived Chief Lower central Upper central Jennings

C1C16 — C2×SD64
C1C2C4C8C16C2×C16C2×D16 — C2×SD64
C1C2C4C8C16 — C2×SD64
C1C22C2×C4C2×C8C2×C16 — C2×SD64
C1C2C2C2C2C2C2C2C2C4C4C4C4C8C8C16 — C2×SD64

Generators and relations for C2×SD64
 G = < a,b,c | a2=b32=c2=1, ab=ba, ac=ca, cbc=b15 >

16C2
16C2
8C4
8C22
8C4
8C22
16C22
16C22
4Q8
4D4
4D4
4Q8
8D4
8C2×C4
8Q8
8C23
2Q16
2D8
2Q16
2D8
4D8
4Q16
4C2×D4
4C2×Q8
2C2×Q16
2Q32
2C2×D8
2D16

Smallest permutation representation of C2×SD64
On 64 points
Generators in S64
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)
(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 55)(2 38)(3 53)(4 36)(5 51)(6 34)(7 49)(8 64)(9 47)(10 62)(11 45)(12 60)(13 43)(14 58)(15 41)(16 56)(17 39)(18 54)(19 37)(20 52)(21 35)(22 50)(23 33)(24 48)(25 63)(26 46)(27 61)(28 44)(29 59)(30 42)(31 57)(32 40)

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim111112222222
type+++++++++++
imageC1C2C2C2C2D4D4D8D8D16D16SD64
kernelC2×SD64C2×C32SD64C2×D16C2×Q32C16C2×C8C8C2×C4C4C22C2
# reps1141111224416

Matrix representation of C2×SD64 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] >;

C2×SD64 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 C2×SD64 in TeX

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