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 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
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