p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.47C23, C4.572+ 1+4, (D4×Q8).6C2, C4⋊C4.156D4, C8⋊9D4.1C2, Q8.Q8⋊35C2, C4⋊2Q16⋊38C2, (C2×D4).316D4, C8.D4⋊24C2, (C2×C8).97C23, C22⋊C4.49D4, C2.48(Q8○D8), Q16⋊C4⋊21C2, C8.18D4⋊37C2, C4⋊C4.234C23, C4⋊C8.101C22, (C2×C4).504C24, Q8.22(C4○D4), C22⋊Q16⋊30C2, C23.323(C2×D4), C4⋊Q8.149C22, C8⋊C4.42C22, C4.Q8.55C22, C2.D8.59C22, (C4×D4).157C22, C22⋊C8.79C22, (C4×Q8).156C22, (C2×Q16).83C22, (C2×Q8).217C23, C2.140(D4⋊5D4), C22⋊Q8.80C22, C23.38D4⋊12C2, C23.47D4⋊16C2, C23.20D4⋊33C2, (C22×C8).307C22, Q8⋊C4.70C22, C22.764(C22×D4), C22.4(C8.C22), C42.C2.39C22, C42.30C22⋊9C2, (C22×C4).1148C23, (C22×Q8).342C22, C42⋊C2.188C22, (C2×M4(2)).112C22, C22.46C24.1C2, C4.229(C2×C4○D4), (C2×C4).601(C2×D4), (C2×Q8⋊C4)⋊32C2, C2.74(C2×C8.C22), (C2×C4⋊C4).667C22, SmallGroup(128,2044)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.47C23
G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, ede-1=b2d >
Subgroups: 336 in 188 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×Q16, C22×Q8, C2×Q8⋊C4, C23.38D4, C8⋊9D4, Q16⋊C4, C22⋊Q16, C4⋊2Q16, C8.18D4, C8.D4, Q8.Q8, C23.47D4, C23.20D4, C42.30C22, D4×Q8, C22.46C24, C42.47C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C8.C22, Q8○D8, C42.47C23
Character table of C42.47C23
| 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 | 2 | 2 | 4 | 2 | 2 | 4 | 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 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 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 | 2 | 0 | 0 | -2 | 2i | 0 | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | -2i | 0 | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 2i | 0 | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | -2i | 0 | 2i | -2 | 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 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | -4 | -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 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | -4 | -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 | 0 | 0 | symplectic lifted from C8.C22, 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 | 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 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
►On 64 points
(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 26 23)(2 20 27 24)(3 17 28 21)(4 18 25 22)(5 16 12 62)(6 13 9 63)(7 14 10 64)(8 15 11 61)(29 43 33 39)(30 44 34 40)(31 41 35 37)(32 42 36 38)(45 57 52 54)(46 58 49 55)(47 59 50 56)(48 60 51 53)
(1 55 28 60)(2 57 25 56)(3 53 26 58)(4 59 27 54)(5 40 10 42)(6 43 11 37)(7 38 12 44)(8 41 9 39)(13 29 61 35)(14 36 62 30)(15 31 63 33)(16 34 64 32)(17 51 23 46)(18 47 24 52)(19 49 21 48)(20 45 22 50)
(1 38 26 42)(2 43 27 39)(3 40 28 44)(4 41 25 37)(5 49 12 46)(6 47 9 50)(7 51 10 48)(8 45 11 52)(13 56 63 59)(14 60 64 53)(15 54 61 57)(16 58 62 55)(17 34 21 30)(18 31 22 35)(19 36 23 32)(20 29 24 33)
(1 23 26 19)(2 18 27 22)(3 21 28 17)(4 20 25 24)(5 64 12 14)(6 13 9 63)(7 62 10 16)(8 15 11 61)(29 41 33 37)(30 40 34 44)(31 43 35 39)(32 38 36 42)(45 57 52 54)(46 53 49 60)(47 59 50 56)(48 55 51 58)
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), (1,19,26,23)(2,20,27,24)(3,17,28,21)(4,18,25,22)(5,16,12,62)(6,13,9,63)(7,14,10,64)(8,15,11,61)(29,43,33,39)(30,44,34,40)(31,41,35,37)(32,42,36,38)(45,57,52,54)(46,58,49,55)(47,59,50,56)(48,60,51,53), (1,55,28,60)(2,57,25,56)(3,53,26,58)(4,59,27,54)(5,40,10,42)(6,43,11,37)(7,38,12,44)(8,41,9,39)(13,29,61,35)(14,36,62,30)(15,31,63,33)(16,34,64,32)(17,51,23,46)(18,47,24,52)(19,49,21,48)(20,45,22,50), (1,38,26,42)(2,43,27,39)(3,40,28,44)(4,41,25,37)(5,49,12,46)(6,47,9,50)(7,51,10,48)(8,45,11,52)(13,56,63,59)(14,60,64,53)(15,54,61,57)(16,58,62,55)(17,34,21,30)(18,31,22,35)(19,36,23,32)(20,29,24,33), (1,23,26,19)(2,18,27,22)(3,21,28,17)(4,20,25,24)(5,64,12,14)(6,13,9,63)(7,62,10,16)(8,15,11,61)(29,41,33,37)(30,40,34,44)(31,43,35,39)(32,38,36,42)(45,57,52,54)(46,53,49,60)(47,59,50,56)(48,55,51,58)>;
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), (1,19,26,23)(2,20,27,24)(3,17,28,21)(4,18,25,22)(5,16,12,62)(6,13,9,63)(7,14,10,64)(8,15,11,61)(29,43,33,39)(30,44,34,40)(31,41,35,37)(32,42,36,38)(45,57,52,54)(46,58,49,55)(47,59,50,56)(48,60,51,53), (1,55,28,60)(2,57,25,56)(3,53,26,58)(4,59,27,54)(5,40,10,42)(6,43,11,37)(7,38,12,44)(8,41,9,39)(13,29,61,35)(14,36,62,30)(15,31,63,33)(16,34,64,32)(17,51,23,46)(18,47,24,52)(19,49,21,48)(20,45,22,50), (1,38,26,42)(2,43,27,39)(3,40,28,44)(4,41,25,37)(5,49,12,46)(6,47,9,50)(7,51,10,48)(8,45,11,52)(13,56,63,59)(14,60,64,53)(15,54,61,57)(16,58,62,55)(17,34,21,30)(18,31,22,35)(19,36,23,32)(20,29,24,33), (1,23,26,19)(2,18,27,22)(3,21,28,17)(4,20,25,24)(5,64,12,14)(6,13,9,63)(7,62,10,16)(8,15,11,61)(29,41,33,37)(30,40,34,44)(31,43,35,39)(32,38,36,42)(45,57,52,54)(46,53,49,60)(47,59,50,56)(48,55,51,58) );
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)], [(1,19,26,23),(2,20,27,24),(3,17,28,21),(4,18,25,22),(5,16,12,62),(6,13,9,63),(7,14,10,64),(8,15,11,61),(29,43,33,39),(30,44,34,40),(31,41,35,37),(32,42,36,38),(45,57,52,54),(46,58,49,55),(47,59,50,56),(48,60,51,53)], [(1,55,28,60),(2,57,25,56),(3,53,26,58),(4,59,27,54),(5,40,10,42),(6,43,11,37),(7,38,12,44),(8,41,9,39),(13,29,61,35),(14,36,62,30),(15,31,63,33),(16,34,64,32),(17,51,23,46),(18,47,24,52),(19,49,21,48),(20,45,22,50)], [(1,38,26,42),(2,43,27,39),(3,40,28,44),(4,41,25,37),(5,49,12,46),(6,47,9,50),(7,51,10,48),(8,45,11,52),(13,56,63,59),(14,60,64,53),(15,54,61,57),(16,58,62,55),(17,34,21,30),(18,31,22,35),(19,36,23,32),(20,29,24,33)], [(1,23,26,19),(2,18,27,22),(3,21,28,17),(4,20,25,24),(5,64,12,14),(6,13,9,63),(7,62,10,16),(8,15,11,61),(29,41,33,37),(30,40,34,44),(31,43,35,39),(32,38,36,42),(45,57,52,54),(46,53,49,60),(47,59,50,56),(48,55,51,58)]])
Matrix representation of C42.47C23 ►in GL6(𝔽17)
| 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 16 |
| 0 | 0 | 7 | 7 | 1 | 0 |
| 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 | 7 | 8 | 0 | 16 |
| 0 | 0 | 9 | 7 | 1 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 3 | 15 | 0 |
| 0 | 0 | 0 | 15 | 0 | 15 |
| 0 | 0 | 0 | 7 | 16 | 14 |
| 0 | 0 | 0 | 10 | 0 | 2 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 2 | 7 | 0 | 13 |
| 0 | 0 | 2 | 10 | 13 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 16 |
| 0 | 0 | 7 | 7 | 1 | 0 |
G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,10,7,0,0,16,0,7,7,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,7,9,0,0,1,0,8,7,0,0,0,0,0,1,0,0,0,0,16,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,3,15,7,10,0,0,15,0,16,0,0,0,0,15,14,2],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,2,2,0,0,0,13,7,10,0,0,0,0,0,13,0,0,0,0,13,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,10,7,0,0,16,0,7,7,0,0,0,0,0,1,0,0,0,0,16,0] >;
C42.47C23 in GAP, Magma, Sage, TeX
C_4^2._{47}C_2^3 % in TeX
G:=Group("C4^2.47C2^3"); // GroupNames label
G:=SmallGroup(128,2044);
// by ID
G=gap.SmallGroup(128,2044);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,723,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2*b^2,d^2=e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,e*d*e^-1=b^2*d>;
// generators/relations
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