p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.508C23, C4.292- 1+4, (D4×Q8)⋊11C2, C8⋊4Q8⋊4C2, C4⋊C4.173D4, (C4×D8).20C2, D4⋊Q8⋊42C2, C4.Q16⋊42C2, (C2×Q8).241D4, C2.63(Q8○D8), D4.36(C4○D4), D4.D4⋊25C2, C4⋊C8.132C22, C4⋊C4.435C23, C4.51(C8⋊C22), (C4×C8).231C22, (C2×C8).115C23, (C2×C4).559C24, C4.SD16⋊22C2, D4.2D4.2C2, C4⋊Q8.188C22, SD16⋊C4⋊43C2, C8⋊C4.58C22, C2.67(Q8⋊5D4), (C2×D8).148C22, (C2×D4).430C23, (C4×D4).198C22, (C2×Q8).255C23, (C4×Q8).190C22, C2.D8.205C22, Q8⋊C4.86C22, (C2×SD16).67C22, C4.4D4.78C22, C22.819(C22×D4), D4⋊C4.209C22, C42.28C22⋊20C2, C22.50C24⋊11C2, C4.260(C2×C4○D4), (C2×C4).635(C2×D4), C2.87(C2×C8⋊C22), SmallGroup(128,2099)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.508C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=b2d >
Subgroups: 352 in 186 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C2.D8, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C22⋊Q8, C4.4D4, C42⋊2C2, C4⋊Q8, C4⋊Q8, C2×D8, C2×SD16, C22×Q8, C4×D8, SD16⋊C4, C8⋊4Q8, D4.D4, D4.2D4, D4⋊Q8, C4.Q16, C4.SD16, C42.28C22, D4×Q8, C22.50C24, C42.508C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C2×C8⋊C22, Q8○D8, C42.508C23
Character table of C42.508C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
►On 64 points
Generators in S64
(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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(5 13)(6 14)(7 15)(8 16)(9 62)(10 63)(11 64)(12 61)(17 23)(18 24)(19 21)(20 22)(29 37)(30 38)(31 39)(32 40)(33 43)(34 44)(35 41)(36 42)(45 49)(46 50)(47 51)(48 52)(53 55)(54 56)(57 59)(58 60)
(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)
(1 29)(2 30)(3 31)(4 32)(5 56)(6 53)(7 54)(8 55)(9 60)(10 57)(11 58)(12 59)(13 46)(14 47)(15 48)(16 45)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(49 63)(50 64)(51 61)(52 62)
G:=sub<Sym(64)| (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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,23)(18,24)(19,21)(20,22)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,41)(36,42)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,29)(2,30)(3,31)(4,32)(5,56)(6,53)(7,54)(8,55)(9,60)(10,57)(11,58)(12,59)(13,46)(14,47)(15,48)(16,45)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(49,63)(50,64)(51,61)(52,62)>;
G:=Group( (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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (5,13)(6,14)(7,15)(8,16)(9,62)(10,63)(11,64)(12,61)(17,23)(18,24)(19,21)(20,22)(29,37)(30,38)(31,39)(32,40)(33,43)(34,44)(35,41)(36,42)(45,49)(46,50)(47,51)(48,52)(53,55)(54,56)(57,59)(58,60), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,29)(2,30)(3,31)(4,32)(5,56)(6,53)(7,54)(8,55)(9,60)(10,57)(11,58)(12,59)(13,46)(14,47)(15,48)(16,45)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(49,63)(50,64)(51,61)(52,62) );
G=PermutationGroup([[(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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(5,13),(6,14),(7,15),(8,16),(9,62),(10,63),(11,64),(12,61),(17,23),(18,24),(19,21),(20,22),(29,37),(30,38),(31,39),(32,40),(33,43),(34,44),(35,41),(36,42),(45,49),(46,50),(47,51),(48,52),(53,55),(54,56),(57,59),(58,60)], [(1,57,27,55),(2,60,28,54),(3,59,25,53),(4,58,26,56),(5,44,11,30),(6,43,12,29),(7,42,9,32),(8,41,10,31),(13,36,64,40),(14,35,61,39),(15,34,62,38),(16,33,63,37),(17,51,21,47),(18,50,22,46),(19,49,23,45),(20,52,24,48)], [(1,29),(2,30),(3,31),(4,32),(5,56),(6,53),(7,54),(8,55),(9,60),(10,57),(11,58),(12,59),(13,46),(14,47),(15,48),(16,45),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,41),(26,42),(27,43),(28,44),(49,63),(50,64),(51,61),(52,62)]])
Matrix representation of C42.508C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 13 | 0 |
| 0 | 0 | 4 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 15 | 0 |
| 0 | 0 | 1 | 0 | 0 | 15 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 3 | 14 | 0 | 0 |
| 0 | 0 | 3 | 0 | 14 | 14 |
| 0 | 0 | 0 | 3 | 14 | 3 |
G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,4,0,0,0,4,4,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,16,0,0,0,1,0,0,1,0,0,0,0,0,16,0,0,0,0,1,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,15,0,0,16,0,0,0,15,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,3,3,3,0,0,0,3,14,0,3,0,0,0,0,14,14,0,0,0,0,14,3] >;
C42.508C23 in GAP, Magma, Sage, TeX
C_4^2._{508}C_2^3 % in TeX
G:=Group("C4^2.508C2^3"); // GroupNames label
G:=SmallGroup(128,2099);
// by ID
G=gap.SmallGroup(128,2099);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,723,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,e*d*e=b^2*d>;
// generators/relations
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