p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.516C23, C4.372- 1+4, Q82⋊5C2, C4⋊C4.181D4, (C4×Q16)⋊45C2, C8⋊4Q8⋊11C2, D4⋊Q8⋊45C2, C4.Q16⋊44C2, C2.65(D4○D8), (C2×Q8).247D4, C4⋊SD16.2C2, C4⋊C4.441C23, C4⋊C8.140C22, (C2×C4).567C24, (C2×C8).121C23, (C4×C8).234C22, Q8.39(C4○D4), Q8.D4⋊47C2, C4.4D8.10C2, C4⋊Q8.196C22, SD16⋊C4⋊47C2, C8⋊C4.66C22, C2.75(Q8⋊5D4), (C2×D4).276C23, (C4×D4).205C22, C4.81(C8.C22), (C2×Q8).406C23, (C4×Q8).198C22, C2.D8.208C22, D4⋊C4.90C22, C4⋊1D4.103C22, (C2×Q16).144C22, (C2×SD16).72C22, C4.4D4.83C22, C22.827(C22×D4), Q8⋊C4.211C22, C42.28C22⋊24C2, C22.53C24.5C2, C4.268(C2×C4○D4), (C2×C4).643(C2×D4), C2.89(C2×C8.C22), SmallGroup(128,2107)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.516C23
G = < a,b,c,d,e | a4=b4=1, c2=e2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >
Subgroups: 336 in 175 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C2.D8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C4×Q16, SD16⋊C4, C8⋊4Q8, C4⋊SD16, Q8.D4, D4⋊Q8, C4.Q16, C4.4D8, C42.28C22, Q82, C22.53C24, C42.516C23
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○D8, C42.516C23
Character table of C42.516C23
| 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 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 2 | 2 | 0 | 2 | -2 | 0 | 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 | -2 | -2 | 0 | 2 | 2 | 0 | 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 | -2 | 2 | 0 | -2 | 2 | 0 | 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 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2 | 2i | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2 | -2i | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 2 | -2i | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | -2 | 2i | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 | orthogonal lifted from D4○D8 |
| ρ26 | 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 | orthogonal lifted from D4○D8 |
| ρ27 | 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 2- 1+4, Schur index 2 |
| ρ28 | 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 C8.C22, Schur index 2 |
| ρ29 | 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 C8.C22, 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 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 35 37 41)(30 36 38 42)(31 33 39 43)(32 34 40 44)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(1 55 27 59)(2 56 28 60)(3 53 25 57)(4 54 26 58)(5 36 9 42)(6 33 10 43)(7 34 11 44)(8 35 12 41)(13 32 62 40)(14 29 63 37)(15 30 64 38)(16 31 61 39)(17 51 21 45)(18 52 22 46)(19 49 23 47)(20 50 24 48)
(1 57 25 55)(2 60 26 54)(3 59 27 53)(4 58 28 56)(5 32 11 38)(6 31 12 37)(7 30 9 40)(8 29 10 39)(13 36 64 44)(14 35 61 43)(15 34 62 42)(16 33 63 41)(17 49 23 45)(18 52 24 48)(19 51 21 47)(20 50 22 46)
(1 29 27 37)(2 30 28 38)(3 31 25 39)(4 32 26 40)(5 56 9 60)(6 53 10 57)(7 54 11 58)(8 55 12 59)(13 46 62 52)(14 47 63 49)(15 48 64 50)(16 45 61 51)(17 33 21 43)(18 34 22 44)(19 35 23 41)(20 36 24 42)
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), (1,23,27,19)(2,24,28,20)(3,21,25,17)(4,22,26,18)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,35,37,41)(30,36,38,42)(31,33,39,43)(32,34,40,44)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,55,27,59)(2,56,28,60)(3,53,25,57)(4,54,26,58)(5,36,9,42)(6,33,10,43)(7,34,11,44)(8,35,12,41)(13,32,62,40)(14,29,63,37)(15,30,64,38)(16,31,61,39)(17,51,21,45)(18,52,22,46)(19,49,23,47)(20,50,24,48), (1,57,25,55)(2,60,26,54)(3,59,27,53)(4,58,28,56)(5,32,11,38)(6,31,12,37)(7,30,9,40)(8,29,10,39)(13,36,64,44)(14,35,61,43)(15,34,62,42)(16,33,63,41)(17,49,23,45)(18,52,24,48)(19,51,21,47)(20,50,22,46), (1,29,27,37)(2,30,28,38)(3,31,25,39)(4,32,26,40)(5,56,9,60)(6,53,10,57)(7,54,11,58)(8,55,12,59)(13,46,62,52)(14,47,63,49)(15,48,64,50)(16,45,61,51)(17,33,21,43)(18,34,22,44)(19,35,23,41)(20,36,24,42)>;
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), (1,23,27,19)(2,24,28,20)(3,21,25,17)(4,22,26,18)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,35,37,41)(30,36,38,42)(31,33,39,43)(32,34,40,44)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,55,27,59)(2,56,28,60)(3,53,25,57)(4,54,26,58)(5,36,9,42)(6,33,10,43)(7,34,11,44)(8,35,12,41)(13,32,62,40)(14,29,63,37)(15,30,64,38)(16,31,61,39)(17,51,21,45)(18,52,22,46)(19,49,23,47)(20,50,24,48), (1,57,25,55)(2,60,26,54)(3,59,27,53)(4,58,28,56)(5,32,11,38)(6,31,12,37)(7,30,9,40)(8,29,10,39)(13,36,64,44)(14,35,61,43)(15,34,62,42)(16,33,63,41)(17,49,23,45)(18,52,24,48)(19,51,21,47)(20,50,22,46), (1,29,27,37)(2,30,28,38)(3,31,25,39)(4,32,26,40)(5,56,9,60)(6,53,10,57)(7,54,11,58)(8,55,12,59)(13,46,62,52)(14,47,63,49)(15,48,64,50)(16,45,61,51)(17,33,21,43)(18,34,22,44)(19,35,23,41)(20,36,24,42) );
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)], [(1,23,27,19),(2,24,28,20),(3,21,25,17),(4,22,26,18),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,35,37,41),(30,36,38,42),(31,33,39,43),(32,34,40,44),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(1,55,27,59),(2,56,28,60),(3,53,25,57),(4,54,26,58),(5,36,9,42),(6,33,10,43),(7,34,11,44),(8,35,12,41),(13,32,62,40),(14,29,63,37),(15,30,64,38),(16,31,61,39),(17,51,21,45),(18,52,22,46),(19,49,23,47),(20,50,24,48)], [(1,57,25,55),(2,60,26,54),(3,59,27,53),(4,58,28,56),(5,32,11,38),(6,31,12,37),(7,30,9,40),(8,29,10,39),(13,36,64,44),(14,35,61,43),(15,34,62,42),(16,33,63,41),(17,49,23,45),(18,52,24,48),(19,51,21,47),(20,50,22,46)], [(1,29,27,37),(2,30,28,38),(3,31,25,39),(4,32,26,40),(5,56,9,60),(6,53,10,57),(7,54,11,58),(8,55,12,59),(13,46,62,52),(14,47,63,49),(15,48,64,50),(16,45,61,51),(17,33,21,43),(18,34,22,44),(19,35,23,41),(20,36,24,42)]])
Matrix representation of C42.516C23 ►in GL6(𝔽17)
| 16 | 15 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 4 | 8 | 0 | 0 | 0 | 0 |
| 13 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 15 | 2 |
| 0 | 0 | 0 | 12 | 15 | 15 |
| 0 | 0 | 15 | 15 | 0 | 5 |
| 0 | 0 | 2 | 15 | 5 | 0 |
| 4 | 8 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 2 | 15 |
| 0 | 0 | 0 | 12 | 15 | 15 |
| 0 | 0 | 2 | 15 | 5 | 0 |
| 0 | 0 | 15 | 15 | 0 | 5 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,5,0,15,2,0,0,0,12,15,15,0,0,15,15,0,5,0,0,2,15,5,0],[4,0,0,0,0,0,8,13,0,0,0,0,0,0,12,0,2,15,0,0,0,12,15,15,0,0,2,15,5,0,0,0,15,15,0,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.516C23 in GAP, Magma, Sage, TeX
C_4^2._{516}C_2^3 % in TeX
G:=Group("C4^2.516C2^3"); // GroupNames label
G:=SmallGroup(128,2107);
// by ID
G=gap.SmallGroup(128,2107);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=e^2=b^2,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^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations
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