direct product, p-group, metabelian, nilpotent (class 4), monomial
Aliases: C22×SD32, C16⋊3C23, C8.10C24, Q16⋊1C23, D8.1C23, C23.64D8, (C2×C4).95D8, C4.22(C2×D8), C8.55(C2×D4), (C2×C8).263D4, (C22×C16)⋊12C2, (C2×C16)⋊20C22, C2.25(C22×D8), C22.76(C2×D8), C4.16(C22×D4), (C2×C8).572C23, (C22×Q16)⋊14C2, (C2×Q16)⋊49C22, (C22×D8).10C2, (C22×C4).622D4, (C2×D8).150C22, (C22×C8).542C22, (C2×C4).873(C2×D4), SmallGroup(128,2141)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C22×SD32
G = < a,b,c,d | a2=b2=c16=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
Subgroups: 500 in 200 conjugacy classes, 100 normal (11 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C16, C2×C8, D8, D8, Q16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2×C16, SD32, C22×C8, C2×D8, C2×D8, C2×Q16, C2×Q16, C22×D4, C22×Q8, C22×C16, C2×SD32, C22×D8, C22×Q16, C22×SD32
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, C24, SD32, C2×D8, C22×D4, C2×SD32, C22×D8, C22×SD32
►On 64 points
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 33)(26 34)(27 35)(28 36)(29 37)(30 38)(31 39)(32 40)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 33)(10 34)(11 35)(12 36)(13 37)(14 38)(15 39)(16 40)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)
(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 41)(2 48)(3 39)(4 46)(5 37)(6 44)(7 35)(8 42)(9 33)(10 40)(11 47)(12 38)(13 45)(14 36)(15 43)(16 34)(17 57)(18 64)(19 55)(20 62)(21 53)(22 60)(23 51)(24 58)(25 49)(26 56)(27 63)(28 54)(29 61)(30 52)(31 59)(32 50)
G:=sub<Sym(64)| (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56), (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), (1,41)(2,48)(3,39)(4,46)(5,37)(6,44)(7,35)(8,42)(9,33)(10,40)(11,47)(12,38)(13,45)(14,36)(15,43)(16,34)(17,57)(18,64)(19,55)(20,62)(21,53)(22,60)(23,51)(24,58)(25,49)(26,56)(27,63)(28,54)(29,61)(30,52)(31,59)(32,50)>;
G:=Group( (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(31,39)(32,40), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,33)(10,34)(11,35)(12,36)(13,37)(14,38)(15,39)(16,40)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56), (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), (1,41)(2,48)(3,39)(4,46)(5,37)(6,44)(7,35)(8,42)(9,33)(10,40)(11,47)(12,38)(13,45)(14,36)(15,43)(16,34)(17,57)(18,64)(19,55)(20,62)(21,53)(22,60)(23,51)(24,58)(25,49)(26,56)(27,63)(28,54)(29,61)(30,52)(31,59)(32,50) );
G=PermutationGroup([[(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,33),(26,34),(27,35),(28,36),(29,37),(30,38),(31,39),(32,40)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,33),(10,34),(11,35),(12,36),(13,37),(14,38),(15,39),(16,40),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56)], [(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)], [(1,41),(2,48),(3,39),(4,46),(5,37),(6,44),(7,35),(8,42),(9,33),(10,40),(11,47),(12,38),(13,45),(14,36),(15,43),(16,34),(17,57),(18,64),(19,55),(20,62),(21,53),(22,60),(23,51),(24,58),(25,49),(26,56),(27,63),(28,54),(29,61),(30,52),(31,59),(32,50)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 8A | ··· | 8H | 16A | ··· | 16P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 16 | ··· | 16 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D8 | D8 | SD32 |
| kernel | C22×SD32 | C22×C16 | C2×SD32 | C22×D8 | C22×Q16 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 3 | 1 | 6 | 2 | 16 |
Matrix representation of C22×SD32 ►in GL4(𝔽17) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 16 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 16 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,7,16,0,0,1,7],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;
C22×SD32 in GAP, Magma, Sage, TeX
C_2^2\times {\rm SD}_{32} % in TeX
G:=Group("C2^2xSD32"); // GroupNames label
G:=SmallGroup(128,2141);
// by ID
G=gap.SmallGroup(128,2141);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,-2,-2,448,253,1684,851,242,4037,2028,124]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^16=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations