p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.58C23, C4.792- 1+4, C8⋊Q8⋊24C2, C4⋊C4.166D4, C8⋊9D4.2C2, Q8.Q8⋊40C2, Q8⋊Q8⋊21C2, (C2×D4).330D4, C8.32(C4○D4), C8.D4⋊31C2, D4⋊3Q8.6C2, Q16⋊C4⋊25C2, C8.18D4⋊31C2, C4⋊C8.117C22, C4⋊C4.249C23, (C2×C8).364C23, (C2×C4).536C24, C22⋊C4.176D4, C23.481(C2×D4), C4⋊Q8.168C22, C2.89(D4⋊6D4), C8⋊C4.50C22, C4.Q8.66C22, C2.89(D4○SD16), (C4×D4).176C22, C22⋊C8.95C22, (C4×Q8).177C22, (C2×Q16).88C22, (C2×Q8).240C23, M4(2)⋊C4⋊31C2, C2.D8.129C22, C23.48D4⋊32C2, C23.20D4⋊43C2, C23.47D4⋊21C2, (C22×C4).340C23, (C22×C8).287C22, Q8⋊C4.77C22, C22.796(C22×D4), C22.6(C8.C22), C22⋊Q8.103C22, C42.C2.49C22, C42⋊C2.207C22, (C2×M4(2)).129C22, C22.46C24.3C2, (C2×C4.Q8)⋊12C2, C4.118(C2×C4○D4), (C2×C4).620(C2×D4), C2.82(C2×C8.C22), (C2×C4⋊C4).685C22, SmallGroup(128,2076)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.58C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=a2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >
Subgroups: 296 in 172 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, D4, 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, C22×C8, C2×M4(2), C2×Q16, C2×C4.Q8, M4(2)⋊C4, C8⋊9D4, Q16⋊C4, C8.18D4, C8.D4, Q8⋊Q8, Q8.Q8, C23.47D4, C23.48D4, C23.20D4, C8⋊Q8, C22.46C24, D4⋊3Q8, C42.58C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, C2×C8.C22, D4○SD16, C42.58C23
Character table of C42.58C23
| 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 | 0 | -2 | -2 | 0 | 2 | 2 | 0 | 0 | 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 | 0 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 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 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 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 | 0 | 2 | 2 | 0 | 2 | -2 | 0 | 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 | -2 | -2i | 0 | 0 | -2i | 0 | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2i | 0 | 0 | -2i | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | -2i | 0 | 0 | 2i | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 2i | 0 | 0 | 2i | 0 | 0 | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 2 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| 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 | -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
(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 27 21)(2 17 28 22)(3 18 25 23)(4 19 26 24)(5 14 63 11)(6 15 64 12)(7 16 61 9)(8 13 62 10)(29 35 44 38)(30 36 41 39)(31 33 42 40)(32 34 43 37)(45 52 57 54)(46 49 58 55)(47 50 59 56)(48 51 60 53)
(1 59 25 45)(2 58 26 48)(3 57 27 47)(4 60 28 46)(5 44 61 31)(6 43 62 30)(7 42 63 29)(8 41 64 32)(9 40 14 35)(10 39 15 34)(11 38 16 33)(12 37 13 36)(17 49 24 53)(18 52 21 56)(19 51 22 55)(20 50 23 54)
(1 41 3 43)(2 31 4 29)(5 51 7 49)(6 54 8 56)(9 58 11 60)(10 47 12 45)(13 59 15 57)(14 48 16 46)(17 40 19 38)(18 34 20 36)(21 39 23 37)(22 33 24 35)(25 32 27 30)(26 44 28 42)(50 64 52 62)(53 61 55 63)
(2 4)(5 63)(6 62)(7 61)(8 64)(9 16)(10 15)(11 14)(12 13)(17 19)(22 24)(26 28)(29 42)(30 41)(31 44)(32 43)(33 38)(34 37)(35 40)(36 39)(45 47)(50 52)(54 56)(57 59)
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), (1,20,27,21)(2,17,28,22)(3,18,25,23)(4,19,26,24)(5,14,63,11)(6,15,64,12)(7,16,61,9)(8,13,62,10)(29,35,44,38)(30,36,41,39)(31,33,42,40)(32,34,43,37)(45,52,57,54)(46,49,58,55)(47,50,59,56)(48,51,60,53), (1,59,25,45)(2,58,26,48)(3,57,27,47)(4,60,28,46)(5,44,61,31)(6,43,62,30)(7,42,63,29)(8,41,64,32)(9,40,14,35)(10,39,15,34)(11,38,16,33)(12,37,13,36)(17,49,24,53)(18,52,21,56)(19,51,22,55)(20,50,23,54), (1,41,3,43)(2,31,4,29)(5,51,7,49)(6,54,8,56)(9,58,11,60)(10,47,12,45)(13,59,15,57)(14,48,16,46)(17,40,19,38)(18,34,20,36)(21,39,23,37)(22,33,24,35)(25,32,27,30)(26,44,28,42)(50,64,52,62)(53,61,55,63), (2,4)(5,63)(6,62)(7,61)(8,64)(9,16)(10,15)(11,14)(12,13)(17,19)(22,24)(26,28)(29,42)(30,41)(31,44)(32,43)(33,38)(34,37)(35,40)(36,39)(45,47)(50,52)(54,56)(57,59)>;
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), (1,20,27,21)(2,17,28,22)(3,18,25,23)(4,19,26,24)(5,14,63,11)(6,15,64,12)(7,16,61,9)(8,13,62,10)(29,35,44,38)(30,36,41,39)(31,33,42,40)(32,34,43,37)(45,52,57,54)(46,49,58,55)(47,50,59,56)(48,51,60,53), (1,59,25,45)(2,58,26,48)(3,57,27,47)(4,60,28,46)(5,44,61,31)(6,43,62,30)(7,42,63,29)(8,41,64,32)(9,40,14,35)(10,39,15,34)(11,38,16,33)(12,37,13,36)(17,49,24,53)(18,52,21,56)(19,51,22,55)(20,50,23,54), (1,41,3,43)(2,31,4,29)(5,51,7,49)(6,54,8,56)(9,58,11,60)(10,47,12,45)(13,59,15,57)(14,48,16,46)(17,40,19,38)(18,34,20,36)(21,39,23,37)(22,33,24,35)(25,32,27,30)(26,44,28,42)(50,64,52,62)(53,61,55,63), (2,4)(5,63)(6,62)(7,61)(8,64)(9,16)(10,15)(11,14)(12,13)(17,19)(22,24)(26,28)(29,42)(30,41)(31,44)(32,43)(33,38)(34,37)(35,40)(36,39)(45,47)(50,52)(54,56)(57,59) );
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)], [(1,20,27,21),(2,17,28,22),(3,18,25,23),(4,19,26,24),(5,14,63,11),(6,15,64,12),(7,16,61,9),(8,13,62,10),(29,35,44,38),(30,36,41,39),(31,33,42,40),(32,34,43,37),(45,52,57,54),(46,49,58,55),(47,50,59,56),(48,51,60,53)], [(1,59,25,45),(2,58,26,48),(3,57,27,47),(4,60,28,46),(5,44,61,31),(6,43,62,30),(7,42,63,29),(8,41,64,32),(9,40,14,35),(10,39,15,34),(11,38,16,33),(12,37,13,36),(17,49,24,53),(18,52,21,56),(19,51,22,55),(20,50,23,54)], [(1,41,3,43),(2,31,4,29),(5,51,7,49),(6,54,8,56),(9,58,11,60),(10,47,12,45),(13,59,15,57),(14,48,16,46),(17,40,19,38),(18,34,20,36),(21,39,23,37),(22,33,24,35),(25,32,27,30),(26,44,28,42),(50,64,52,62),(53,61,55,63)], [(2,4),(5,63),(6,62),(7,61),(8,64),(9,16),(10,15),(11,14),(12,13),(17,19),(22,24),(26,28),(29,42),(30,41),(31,44),(32,43),(33,38),(34,37),(35,40),(36,39),(45,47),(50,52),(54,56),(57,59)]])
Matrix representation of C42.58C23 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 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 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 16 | 0 | 0 |
| 0 | 0 | 16 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 10 |
| 0 | 0 | 0 | 0 | 10 | 16 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,10,16,0,0,0,0,16,7,0,0,0,0,0,0,1,10,0,0,0,0,10,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;
C42.58C23 in GAP, Magma, Sage, TeX
C_4^2._{58}C_2^3 % in TeX
G:=Group("C4^2.58C2^3"); // GroupNames label
G:=SmallGroup(128,2076);
// by ID
G=gap.SmallGroup(128,2076);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,100,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,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=a^2*c,e*d*e=b^2*d>;
// generators/relations
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