p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.55C23, C4.652+ 1+4, (D4×Q8)⋊8C2, C8⋊D4⋊41C2, C8⋊9D4⋊22C2, C8⋊8D4⋊53C2, C4⋊C4.160D4, Q8.Q8⋊37C2, Q8⋊D4⋊21C2, (C2×D4).320D4, C22⋊C4.53D4, D4.D4⋊22C2, C4⋊C4.238C23, C4⋊C8.106C22, (C2×C4).512C24, (C2×C8).101C23, Q8.26(C4○D4), C23.329(C2×D4), C4⋊Q8.153C22, SD16⋊C4⋊37C2, C8⋊C4.47C22, C2.78(D4○SD16), (C4×D4).163C22, (C2×D4).238C23, C4⋊D4.87C22, C22⋊C8.84C22, (C2×Q8).398C23, (C4×Q8).161C22, C2.148(D4⋊5D4), C4.Q8.106C22, C2.D8.121C22, C22⋊Q8.86C22, C23.38D4⋊14C2, C23.48D4⋊28C2, C23.20D4⋊37C2, (C22×C8).365C22, Q8⋊C4.73C22, (C2×SD16).58C22, C22.772(C22×D4), C22.8(C8.C22), C42.C2.42C22, D4⋊C4.141C22, (C22×C4).1156C23, (C22×Q8).345C22, C42⋊C2.192C22, C42.30C22⋊11C2, (C2×M4(2)).118C22, C22.47C24.1C2, C4.237(C2×C4○D4), (C2×C4).609(C2×D4), (C2×Q8⋊C4)⋊44C2, C2.77(C2×C8.C22), (C2×C4⋊C4).671C22, SmallGroup(128,2052)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.55C23
G = < a,b,c,d,e | a4=b4=1, c2=a2, 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: 376 in 194 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C22×Q8, C2×Q8⋊C4, C23.38D4, C8⋊9D4, SD16⋊C4, Q8⋊D4, D4.D4, C8⋊8D4, C8⋊D4, Q8.Q8, C23.48D4, C23.20D4, C42.30C22, D4×Q8, C22.47C24, C42.55C23
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, D4○SD16, C42.55C23
Character table of C42.55C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 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 | -2 | -2 | -2 | 0 | -2 | -2 | 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 | 0 | -2 | -2 | -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 | 0 | -2 | -2 | -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 | 0 | -2 | -2 | 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 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 2i | 2 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2i | -2 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 2i | -2 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | -2i | 2 | 2i | 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 | 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 | -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 39 23)(2 17 40 24)(3 18 37 21)(4 19 38 22)(5 10 61 29)(6 11 62 30)(7 12 63 31)(8 9 64 32)(13 53 26 58)(14 54 27 59)(15 55 28 60)(16 56 25 57)(33 43 49 46)(34 44 50 47)(35 41 51 48)(36 42 52 45)
(1 16 3 14)(2 28 4 26)(5 49 7 51)(6 36 8 34)(9 47 11 45)(10 43 12 41)(13 40 15 38)(17 55 19 53)(18 59 20 57)(21 54 23 56)(22 58 24 60)(25 37 27 39)(29 46 31 48)(30 42 32 44)(33 63 35 61)(50 62 52 64)
(1 36 39 52)(2 49 40 33)(3 34 37 50)(4 51 38 35)(5 58 61 53)(6 54 62 59)(7 60 63 55)(8 56 64 57)(9 16 32 25)(10 26 29 13)(11 14 30 27)(12 28 31 15)(17 43 24 46)(18 47 21 44)(19 41 22 48)(20 45 23 42)
(1 25 39 16)(2 15 40 28)(3 27 37 14)(4 13 38 26)(5 41 61 48)(6 47 62 44)(7 43 63 46)(8 45 64 42)(9 36 32 52)(10 51 29 35)(11 34 30 50)(12 49 31 33)(17 55 24 60)(18 59 21 54)(19 53 22 58)(20 57 23 56)
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), (1,20,39,23)(2,17,40,24)(3,18,37,21)(4,19,38,22)(5,10,61,29)(6,11,62,30)(7,12,63,31)(8,9,64,32)(13,53,26,58)(14,54,27,59)(15,55,28,60)(16,56,25,57)(33,43,49,46)(34,44,50,47)(35,41,51,48)(36,42,52,45), (1,16,3,14)(2,28,4,26)(5,49,7,51)(6,36,8,34)(9,47,11,45)(10,43,12,41)(13,40,15,38)(17,55,19,53)(18,59,20,57)(21,54,23,56)(22,58,24,60)(25,37,27,39)(29,46,31,48)(30,42,32,44)(33,63,35,61)(50,62,52,64), (1,36,39,52)(2,49,40,33)(3,34,37,50)(4,51,38,35)(5,58,61,53)(6,54,62,59)(7,60,63,55)(8,56,64,57)(9,16,32,25)(10,26,29,13)(11,14,30,27)(12,28,31,15)(17,43,24,46)(18,47,21,44)(19,41,22,48)(20,45,23,42), (1,25,39,16)(2,15,40,28)(3,27,37,14)(4,13,38,26)(5,41,61,48)(6,47,62,44)(7,43,63,46)(8,45,64,42)(9,36,32,52)(10,51,29,35)(11,34,30,50)(12,49,31,33)(17,55,24,60)(18,59,21,54)(19,53,22,58)(20,57,23,56)>;
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), (1,20,39,23)(2,17,40,24)(3,18,37,21)(4,19,38,22)(5,10,61,29)(6,11,62,30)(7,12,63,31)(8,9,64,32)(13,53,26,58)(14,54,27,59)(15,55,28,60)(16,56,25,57)(33,43,49,46)(34,44,50,47)(35,41,51,48)(36,42,52,45), (1,16,3,14)(2,28,4,26)(5,49,7,51)(6,36,8,34)(9,47,11,45)(10,43,12,41)(13,40,15,38)(17,55,19,53)(18,59,20,57)(21,54,23,56)(22,58,24,60)(25,37,27,39)(29,46,31,48)(30,42,32,44)(33,63,35,61)(50,62,52,64), (1,36,39,52)(2,49,40,33)(3,34,37,50)(4,51,38,35)(5,58,61,53)(6,54,62,59)(7,60,63,55)(8,56,64,57)(9,16,32,25)(10,26,29,13)(11,14,30,27)(12,28,31,15)(17,43,24,46)(18,47,21,44)(19,41,22,48)(20,45,23,42), (1,25,39,16)(2,15,40,28)(3,27,37,14)(4,13,38,26)(5,41,61,48)(6,47,62,44)(7,43,63,46)(8,45,64,42)(9,36,32,52)(10,51,29,35)(11,34,30,50)(12,49,31,33)(17,55,24,60)(18,59,21,54)(19,53,22,58)(20,57,23,56) );
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)], [(1,20,39,23),(2,17,40,24),(3,18,37,21),(4,19,38,22),(5,10,61,29),(6,11,62,30),(7,12,63,31),(8,9,64,32),(13,53,26,58),(14,54,27,59),(15,55,28,60),(16,56,25,57),(33,43,49,46),(34,44,50,47),(35,41,51,48),(36,42,52,45)], [(1,16,3,14),(2,28,4,26),(5,49,7,51),(6,36,8,34),(9,47,11,45),(10,43,12,41),(13,40,15,38),(17,55,19,53),(18,59,20,57),(21,54,23,56),(22,58,24,60),(25,37,27,39),(29,46,31,48),(30,42,32,44),(33,63,35,61),(50,62,52,64)], [(1,36,39,52),(2,49,40,33),(3,34,37,50),(4,51,38,35),(5,58,61,53),(6,54,62,59),(7,60,63,55),(8,56,64,57),(9,16,32,25),(10,26,29,13),(11,14,30,27),(12,28,31,15),(17,43,24,46),(18,47,21,44),(19,41,22,48),(20,45,23,42)], [(1,25,39,16),(2,15,40,28),(3,27,37,14),(4,13,38,26),(5,41,61,48),(6,47,62,44),(7,43,63,46),(8,45,64,42),(9,36,32,52),(10,51,29,35),(11,34,30,50),(12,49,31,33),(17,55,24,60),(18,59,21,54),(19,53,22,58),(20,57,23,56)]])
Matrix representation of C42.55C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 6 |
| 0 | 0 | 0 | 4 | 11 | 0 |
| 0 | 0 | 0 | 11 | 13 | 0 |
| 0 | 0 | 6 | 0 | 0 | 13 |
| 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 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 13 | 0 |
| 0 | 0 | 11 | 0 | 0 | 4 |
| 0 | 0 | 13 | 0 | 0 | 11 |
| 0 | 0 | 0 | 4 | 11 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 4 | 14 | 14 |
| 0 | 0 | 4 | 4 | 14 | 3 |
| 0 | 0 | 14 | 14 | 13 | 4 |
| 0 | 0 | 14 | 3 | 4 | 4 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 13 | 0 |
| 0 | 0 | 6 | 0 | 0 | 13 |
| 0 | 0 | 4 | 0 | 0 | 6 |
| 0 | 0 | 0 | 4 | 11 | 0 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,6,0,0,0,4,11,0,0,0,0,11,13,0,0,0,6,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,11,13,0,0,0,11,0,0,4,0,0,13,0,0,11,0,0,0,4,11,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,4,14,14,0,0,4,4,14,3,0,0,14,14,13,4,0,0,14,3,4,4],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,6,4,0,0,0,11,0,0,4,0,0,13,0,0,11,0,0,0,13,6,0] >;
C42.55C23 in GAP, Magma, Sage, TeX
C_4^2._{55}C_2^3 % in TeX
G:=Group("C4^2.55C2^3"); // GroupNames label
G:=SmallGroup(128,2052);
// by ID
G=gap.SmallGroup(128,2052);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,352,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,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
Export