Copied to
clipboard

## 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

 Derived series C1 — C16 — C2×SD64
 Chief series C1 — C2 — C4 — C8 — C16 — C2×C16 — C2×D16 — C2×SD64
 Lower central C1 — C2 — C4 — C8 — C16 — C2×SD64
 Upper central C1 — C22 — C2×C4 — C2×C8 — C2×C16 — C2×SD64
 Jennings C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C16 — C2×SD64

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

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

38 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D8 D8 D16 D16 SD64 kernel C2×SD64 C2×C32 SD64 C2×D16 C2×Q32 C16 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 1 4 1 1 1 1 2 2 4 4 16

Matrix representation of C2×SD64 in GL3(𝔽97) generated by

 96 0 0 0 96 0 0 0 96
,
 1 0 0 0 36 35 0 62 36
,
 1 0 0 0 1 0 0 0 96
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

׿
×
𝔽