p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.513C23, C4.342- 1+4, C8⋊4Q8⋊9C2, C4⋊C4.178D4, Q8⋊3Q8⋊7C2, Q8.Q8⋊48C2, C4.Q16⋊43C2, D4⋊3Q8.8C2, (C2×Q8).244D4, D4.38(C4○D4), C4⋊C4.438C23, C4⋊C8.137C22, (C2×C8).118C23, (C4×C8).299C22, (C2×C4).564C24, C4.SD16⋊37C2, (C4×SD16).17C2, D4.D4.1C2, C4⋊Q8.193C22, C8⋊C4.63C22, C2.72(Q8⋊5D4), SD16⋊C4.1C2, (C2×D4).431C23, (C4×D4).203C22, C4.52(C8.C22), (C2×Q8).258C23, (C4×Q8).195C22, C4.Q8.181C22, C2.D8.136C22, C2.103(D4○SD16), Q8⋊C4.90C22, (C2×SD16).71C22, C22.824(C22×D4), C42.C2.67C22, D4⋊C4.213C22, C42.30C22⋊12C2, C4.265(C2×C4○D4), (C2×C4).640(C2×D4), C2.87(C2×C8.C22), SmallGroup(128,2104)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.513C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=b2d >
Subgroups: 296 in 170 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, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22⋊Q8, C42.C2, C42.C2, C4⋊Q8, C4⋊Q8, C2×SD16, C2×SD16, C4×SD16, SD16⋊C4, C8⋊4Q8, D4.D4, D4.D4, C4.Q16, Q8.Q8, C4.SD16, C42.30C22, D4⋊3Q8, Q8⋊3Q8, C42.513C23
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, D4○SD16, C42.513C23
Character table of C42.513C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | -2 | 2 | -2 | 2 | -2 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | 2 | -2 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 2 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 0 | 2 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | 0 | -4 | 0 | 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 | 0 | 0 | 2√-2 | -2√-2 | 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 | 0 | 0 | -2√-2 | 2√-2 | 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 19 25 21)(2 20 26 22)(3 17 27 23)(4 18 28 24)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 33 39 41)(30 34 40 42)(31 35 37 43)(32 36 38 44)(45 53 57 51)(46 54 58 52)(47 55 59 49)(48 56 60 50)
(1 57 25 45)(2 58 26 46)(3 59 27 47)(4 60 28 48)(5 44 9 36)(6 41 10 33)(7 42 11 34)(8 43 12 35)(13 40 62 30)(14 37 63 31)(15 38 64 32)(16 39 61 29)(17 55 23 49)(18 56 24 50)(19 53 21 51)(20 54 22 52)
(1 33 3 35)(2 36 4 34)(5 58 7 60)(6 57 8 59)(9 46 11 48)(10 45 12 47)(13 50 15 52)(14 49 16 51)(17 37 19 39)(18 40 20 38)(21 29 23 31)(22 32 24 30)(25 41 27 43)(26 44 28 42)(53 63 55 61)(54 62 56 64)
(5 15)(6 16)(7 13)(8 14)(9 64)(10 61)(11 62)(12 63)(17 23)(18 24)(19 21)(20 22)(33 41)(34 42)(35 43)(36 44)(45 51)(46 52)(47 49)(48 50)(53 57)(54 58)(55 59)(56 60)
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), (1,19,25,21)(2,20,26,22)(3,17,27,23)(4,18,28,24)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,33,39,41)(30,34,40,42)(31,35,37,43)(32,36,38,44)(45,53,57,51)(46,54,58,52)(47,55,59,49)(48,56,60,50), (1,57,25,45)(2,58,26,46)(3,59,27,47)(4,60,28,48)(5,44,9,36)(6,41,10,33)(7,42,11,34)(8,43,12,35)(13,40,62,30)(14,37,63,31)(15,38,64,32)(16,39,61,29)(17,55,23,49)(18,56,24,50)(19,53,21,51)(20,54,22,52), (1,33,3,35)(2,36,4,34)(5,58,7,60)(6,57,8,59)(9,46,11,48)(10,45,12,47)(13,50,15,52)(14,49,16,51)(17,37,19,39)(18,40,20,38)(21,29,23,31)(22,32,24,30)(25,41,27,43)(26,44,28,42)(53,63,55,61)(54,62,56,64), (5,15)(6,16)(7,13)(8,14)(9,64)(10,61)(11,62)(12,63)(17,23)(18,24)(19,21)(20,22)(33,41)(34,42)(35,43)(36,44)(45,51)(46,52)(47,49)(48,50)(53,57)(54,58)(55,59)(56,60)>;
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), (1,19,25,21)(2,20,26,22)(3,17,27,23)(4,18,28,24)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,33,39,41)(30,34,40,42)(31,35,37,43)(32,36,38,44)(45,53,57,51)(46,54,58,52)(47,55,59,49)(48,56,60,50), (1,57,25,45)(2,58,26,46)(3,59,27,47)(4,60,28,48)(5,44,9,36)(6,41,10,33)(7,42,11,34)(8,43,12,35)(13,40,62,30)(14,37,63,31)(15,38,64,32)(16,39,61,29)(17,55,23,49)(18,56,24,50)(19,53,21,51)(20,54,22,52), (1,33,3,35)(2,36,4,34)(5,58,7,60)(6,57,8,59)(9,46,11,48)(10,45,12,47)(13,50,15,52)(14,49,16,51)(17,37,19,39)(18,40,20,38)(21,29,23,31)(22,32,24,30)(25,41,27,43)(26,44,28,42)(53,63,55,61)(54,62,56,64), (5,15)(6,16)(7,13)(8,14)(9,64)(10,61)(11,62)(12,63)(17,23)(18,24)(19,21)(20,22)(33,41)(34,42)(35,43)(36,44)(45,51)(46,52)(47,49)(48,50)(53,57)(54,58)(55,59)(56,60) );
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)], [(1,19,25,21),(2,20,26,22),(3,17,27,23),(4,18,28,24),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,33,39,41),(30,34,40,42),(31,35,37,43),(32,36,38,44),(45,53,57,51),(46,54,58,52),(47,55,59,49),(48,56,60,50)], [(1,57,25,45),(2,58,26,46),(3,59,27,47),(4,60,28,48),(5,44,9,36),(6,41,10,33),(7,42,11,34),(8,43,12,35),(13,40,62,30),(14,37,63,31),(15,38,64,32),(16,39,61,29),(17,55,23,49),(18,56,24,50),(19,53,21,51),(20,54,22,52)], [(1,33,3,35),(2,36,4,34),(5,58,7,60),(6,57,8,59),(9,46,11,48),(10,45,12,47),(13,50,15,52),(14,49,16,51),(17,37,19,39),(18,40,20,38),(21,29,23,31),(22,32,24,30),(25,41,27,43),(26,44,28,42),(53,63,55,61),(54,62,56,64)], [(5,15),(6,16),(7,13),(8,14),(9,64),(10,61),(11,62),(12,63),(17,23),(18,24),(19,21),(20,22),(33,41),(34,42),(35,43),(36,44),(45,51),(46,52),(47,49),(48,50),(53,57),(54,58),(55,59),(56,60)]])
Matrix representation of C42.513C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 15 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 9 |
| 0 | 0 | 14 | 11 | 13 | 0 |
| 0 | 0 | 0 | 9 | 6 | 6 |
| 0 | 0 | 13 | 0 | 14 | 11 |
| 7 | 16 | 0 | 0 | 0 | 0 |
| 16 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,6,14,0,13,0,0,6,11,9,0,0,0,0,13,6,14,0,0,9,0,6,11],[7,16,0,0,0,0,16,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,0,1] >;
C42.513C23 in GAP, Magma, Sage, TeX
C_4^2._{513}C_2^3 % in TeX
G:=Group("C4^2.513C2^3"); // GroupNames label
G:=SmallGroup(128,2104);
// by ID
G=gap.SmallGroup(128,2104);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=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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