p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.517C23, C4.382- 1+4, C4⋊C4.182D4, C8⋊4Q8⋊12C2, D4.Q8⋊50C2, Q8.Q8⋊49C2, Q8⋊Q8⋊27C2, D4⋊3Q8⋊14C2, (C4×SD16)⋊64C2, D8⋊C4.1C2, (C2×Q8).138D4, D4.40(C4○D4), D4.D4⋊26C2, C4⋊C4.265C23, C4⋊C8.141C22, (C2×C4).568C24, (C4×C8).301C22, (C2×C8).372C23, D4.2D4.3C2, (C2×D8).94C22, C4⋊Q8.197C22, SD16⋊C4⋊48C2, C8⋊C4.67C22, C2.76(Q8⋊5D4), (C4×D4).206C22, (C2×D4).432C23, (C4×Q8).199C22, (C2×Q8).261C23, C2.D8.137C22, C4.Q8.183C22, C2.105(D4○SD16), D4⋊C4.91C22, (C2×SD16).73C22, C4.4D4.84C22, C22.828(C22×D4), C42.C2.69C22, Q8⋊C4.212C22, C2.103(D8⋊C22), C42.28C22⋊25C2, C42.78C22⋊24C2, C42.30C22⋊13C2, C22.50C24⋊12C2, C4.269(C2×C4○D4), (C2×C4).644(C2×D4), SmallGroup(128,2108)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.517C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, ede=b2d >
Subgroups: 320 in 173 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C2×D8, C2×SD16, C4×SD16, SD16⋊C4, D8⋊C4, C8⋊4Q8, D4.D4, D4.2D4, Q8⋊Q8, D4.Q8, Q8.Q8, C42.78C22, C42.28C22, C42.30C22, D4⋊3Q8, C22.50C24, C42.517C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, D8⋊C22, D4○SD16, C42.517C23
Character table of C42.517C23
| 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 | -2 | -2 | 2 | 0 | 0 | 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 | -2 | 2 | 2 | 0 | 0 | -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 | 2 | -2 | -2 | 0 | 0 | -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 | 2 | 2 | -2 | 0 | 0 | 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 | 0 | 0 | 0 | -2i | 2i | 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 | 0 | 0 | 0 | 2i | -2i | 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 | 0 | 0 | 0 | -2i | 2i | 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 | 0 | 0 | 0 | 2i | -2i | 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 | 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 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| 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 | complex lifted from D4○SD16 |
| 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 | complex lifted from D4○SD16 |
►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 20 26 23)(2 17 27 24)(3 18 28 21)(4 19 25 22)(5 12 63 15)(6 9 64 16)(7 10 61 13)(8 11 62 14)(29 37 41 36)(30 38 42 33)(31 39 43 34)(32 40 44 35)(45 56 60 51)(46 53 57 52)(47 54 58 49)(48 55 59 50)
(1 48 26 59)(2 45 27 60)(3 46 28 57)(4 47 25 58)(5 39 63 34)(6 40 64 35)(7 37 61 36)(8 38 62 33)(9 32 16 44)(10 29 13 41)(11 30 14 42)(12 31 15 43)(17 51 24 56)(18 52 21 53)(19 49 22 54)(20 50 23 55)
(1 45 28 58)(2 59 25 46)(3 47 26 60)(4 57 27 48)(5 44 61 30)(6 31 62 41)(7 42 63 32)(8 29 64 43)(9 39 14 36)(10 33 15 40)(11 37 16 34)(12 35 13 38)(17 50 22 53)(18 54 23 51)(19 52 24 55)(20 56 21 49)
(1 44)(2 41)(3 42)(4 43)(5 47)(6 48)(7 45)(8 46)(9 50)(10 51)(11 52)(12 49)(13 56)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 31)(26 32)(27 29)(28 30)(57 62)(58 63)(59 64)(60 61)
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), (1,20,26,23)(2,17,27,24)(3,18,28,21)(4,19,25,22)(5,12,63,15)(6,9,64,16)(7,10,61,13)(8,11,62,14)(29,37,41,36)(30,38,42,33)(31,39,43,34)(32,40,44,35)(45,56,60,51)(46,53,57,52)(47,54,58,49)(48,55,59,50), (1,48,26,59)(2,45,27,60)(3,46,28,57)(4,47,25,58)(5,39,63,34)(6,40,64,35)(7,37,61,36)(8,38,62,33)(9,32,16,44)(10,29,13,41)(11,30,14,42)(12,31,15,43)(17,51,24,56)(18,52,21,53)(19,49,22,54)(20,50,23,55), (1,45,28,58)(2,59,25,46)(3,47,26,60)(4,57,27,48)(5,44,61,30)(6,31,62,41)(7,42,63,32)(8,29,64,43)(9,39,14,36)(10,33,15,40)(11,37,16,34)(12,35,13,38)(17,50,22,53)(18,54,23,51)(19,52,24,55)(20,56,21,49), (1,44)(2,41)(3,42)(4,43)(5,47)(6,48)(7,45)(8,46)(9,50)(10,51)(11,52)(12,49)(13,56)(14,53)(15,54)(16,55)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,31)(26,32)(27,29)(28,30)(57,62)(58,63)(59,64)(60,61)>;
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), (1,20,26,23)(2,17,27,24)(3,18,28,21)(4,19,25,22)(5,12,63,15)(6,9,64,16)(7,10,61,13)(8,11,62,14)(29,37,41,36)(30,38,42,33)(31,39,43,34)(32,40,44,35)(45,56,60,51)(46,53,57,52)(47,54,58,49)(48,55,59,50), (1,48,26,59)(2,45,27,60)(3,46,28,57)(4,47,25,58)(5,39,63,34)(6,40,64,35)(7,37,61,36)(8,38,62,33)(9,32,16,44)(10,29,13,41)(11,30,14,42)(12,31,15,43)(17,51,24,56)(18,52,21,53)(19,49,22,54)(20,50,23,55), (1,45,28,58)(2,59,25,46)(3,47,26,60)(4,57,27,48)(5,44,61,30)(6,31,62,41)(7,42,63,32)(8,29,64,43)(9,39,14,36)(10,33,15,40)(11,37,16,34)(12,35,13,38)(17,50,22,53)(18,54,23,51)(19,52,24,55)(20,56,21,49), (1,44)(2,41)(3,42)(4,43)(5,47)(6,48)(7,45)(8,46)(9,50)(10,51)(11,52)(12,49)(13,56)(14,53)(15,54)(16,55)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,31)(26,32)(27,29)(28,30)(57,62)(58,63)(59,64)(60,61) );
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)], [(1,20,26,23),(2,17,27,24),(3,18,28,21),(4,19,25,22),(5,12,63,15),(6,9,64,16),(7,10,61,13),(8,11,62,14),(29,37,41,36),(30,38,42,33),(31,39,43,34),(32,40,44,35),(45,56,60,51),(46,53,57,52),(47,54,58,49),(48,55,59,50)], [(1,48,26,59),(2,45,27,60),(3,46,28,57),(4,47,25,58),(5,39,63,34),(6,40,64,35),(7,37,61,36),(8,38,62,33),(9,32,16,44),(10,29,13,41),(11,30,14,42),(12,31,15,43),(17,51,24,56),(18,52,21,53),(19,49,22,54),(20,50,23,55)], [(1,45,28,58),(2,59,25,46),(3,47,26,60),(4,57,27,48),(5,44,61,30),(6,31,62,41),(7,42,63,32),(8,29,64,43),(9,39,14,36),(10,33,15,40),(11,37,16,34),(12,35,13,38),(17,50,22,53),(18,54,23,51),(19,52,24,55),(20,56,21,49)], [(1,44),(2,41),(3,42),(4,43),(5,47),(6,48),(7,45),(8,46),(9,50),(10,51),(11,52),(12,49),(13,56),(14,53),(15,54),(16,55),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,31),(26,32),(27,29),(28,30),(57,62),(58,63),(59,64),(60,61)]])
Matrix representation of C42.517C23 ►in GL6(𝔽17)
| 4 | 2 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 8 | 0 |
| 0 | 0 | 13 | 0 | 4 | 4 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 13 | 4 | 4 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 13 | 0 | 4 | 0 |
| 0 | 0 | 13 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 6 | 6 |
| 0 | 0 | 14 | 0 | 0 | 6 |
| 0 | 0 | 0 | 14 | 3 | 3 |
| 0 | 0 | 14 | 3 | 3 | 3 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,2,13,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,0,0,0,0,0,9,16,0,0,0,0,0,0,13,13,0,13,0,0,0,0,0,4,0,0,8,4,4,4,0,0,0,4,0,0],[4,1,0,0,0,0,0,13,0,0,0,0,0,0,13,0,13,13,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,14,0,14,0,0,0,0,14,3,0,0,6,0,3,3,0,0,6,6,3,3] >;
C42.517C23 in GAP, Magma, Sage, TeX
C_4^2._{517}C_2^3 % in TeX
G:=Group("C4^2.517C2^3"); // GroupNames label
G:=SmallGroup(128,2108);
// by ID
G=gap.SmallGroup(128,2108);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,346,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,e*d*e=b^2*d>;
// generators/relations
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