direct product, p-group, metabelian, nilpotent (class 5), monomial
Aliases: C2×SD64, C4.7D16, C8.19D8, C32⋊3C22, C16.10D4, C16.7C23, Q32⋊1C22, 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
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
8Q8
8C23
2Q16
2D8
2Q16
2D8
4D8
4Q16
2Q32
2D16
Smallest permutation representation of C2×SD64
►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 | 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
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