p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.62C23, C4.832- 1+4, C8⋊Q8⋊28C2, C8⋊D4⋊52C2, C8⋊8D4⋊23C2, C8⋊9D4⋊28C2, C4⋊C4.380D4, D4.Q8⋊43C2, Q8⋊Q8⋊22C2, (C2×D4).180D4, C8.35(C4○D4), C22⋊C4.64D4, C4⋊C4.253C23, C4⋊C8.121C22, (C2×C4).540C24, (C2×C8).107C23, C23.345(C2×D4), C4⋊Q8.172C22, SD16⋊C4⋊41C2, C2.93(D4⋊6D4), C8⋊C4.54C22, C4.Q8.68C22, C2.91(D4○SD16), (C2×D4).258C23, (C4×D4).180C22, C22⋊C8.99C22, (C2×Q8).242C23, (C4×Q8).179C22, M4(2)⋊C4⋊35C2, C2.D8.224C22, D4⋊C4.82C22, C23.20D4⋊45C2, C4⋊D4.107C22, C23.25D4⋊31C2, C23.46D4⋊22C2, C23.19D4⋊45C2, (C22×C8).291C22, Q8⋊C4.79C22, (C2×SD16).65C22, C22.800(C22×D4), C22⋊Q8.105C22, C42.C2.53C22, C2.95(D8⋊C22), (C22×C4).1168C23, C22.46C24⋊10C2, C42⋊C2.211C22, (C2×M4(2)).133C22, C22.49C24.5C2, C4.122(C2×C4○D4), (C2×C4).624(C2×D4), (C2×C4⋊C4).689C22, SmallGroup(128,2080)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.62C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=a2b2, ab=ba, cac-1=eae=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, ede=b2d >
Subgroups: 320 in 174 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, 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, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C23.25D4, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, Q8⋊Q8, D4.Q8, C23.46D4, C23.19D4, C23.20D4, C8⋊Q8, C22.46C24, C22.49C24, C42.62C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, D8⋊C22, D4○SD16, C42.62C23
Character table of C42.62C23
| 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 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 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 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 0 | 2 | 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 | 2i | 0 | -2i | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 4i | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| 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
(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 52 28 31)(2 49 25 32)(3 50 26 29)(4 51 27 30)(5 60 61 54)(6 57 62 55)(7 58 63 56)(8 59 64 53)(9 33 14 40)(10 34 15 37)(11 35 16 38)(12 36 13 39)(17 45 24 41)(18 46 21 42)(19 47 22 43)(20 48 23 44)
(1 56 3 54)(2 57 4 59)(5 31 7 29)(6 51 8 49)(9 43 11 41)(10 46 12 48)(13 44 15 42)(14 47 16 45)(17 40 19 38)(18 36 20 34)(21 39 23 37)(22 35 24 33)(25 55 27 53)(26 60 28 58)(30 64 32 62)(50 61 52 63)
(1 27 26 2)(3 25 28 4)(5 57 63 53)(6 56 64 60)(7 59 61 55)(8 54 62 58)(9 15 16 12)(10 11 13 14)(17 48 22 42)(18 41 23 47)(19 46 24 44)(20 43 21 45)(29 32 52 51)(30 50 49 31)(33 34 38 39)(35 36 40 37)
(1 16)(2 10)(3 14)(4 12)(5 17)(6 23)(7 19)(8 21)(9 26)(11 28)(13 27)(15 25)(18 64)(20 62)(22 63)(24 61)(29 33)(30 39)(31 35)(32 37)(34 49)(36 51)(38 52)(40 50)(41 54)(42 59)(43 56)(44 57)(45 60)(46 53)(47 58)(48 55)
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), (1,52,28,31)(2,49,25,32)(3,50,26,29)(4,51,27,30)(5,60,61,54)(6,57,62,55)(7,58,63,56)(8,59,64,53)(9,33,14,40)(10,34,15,37)(11,35,16,38)(12,36,13,39)(17,45,24,41)(18,46,21,42)(19,47,22,43)(20,48,23,44), (1,56,3,54)(2,57,4,59)(5,31,7,29)(6,51,8,49)(9,43,11,41)(10,46,12,48)(13,44,15,42)(14,47,16,45)(17,40,19,38)(18,36,20,34)(21,39,23,37)(22,35,24,33)(25,55,27,53)(26,60,28,58)(30,64,32,62)(50,61,52,63), (1,27,26,2)(3,25,28,4)(5,57,63,53)(6,56,64,60)(7,59,61,55)(8,54,62,58)(9,15,16,12)(10,11,13,14)(17,48,22,42)(18,41,23,47)(19,46,24,44)(20,43,21,45)(29,32,52,51)(30,50,49,31)(33,34,38,39)(35,36,40,37), (1,16)(2,10)(3,14)(4,12)(5,17)(6,23)(7,19)(8,21)(9,26)(11,28)(13,27)(15,25)(18,64)(20,62)(22,63)(24,61)(29,33)(30,39)(31,35)(32,37)(34,49)(36,51)(38,52)(40,50)(41,54)(42,59)(43,56)(44,57)(45,60)(46,53)(47,58)(48,55)>;
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), (1,52,28,31)(2,49,25,32)(3,50,26,29)(4,51,27,30)(5,60,61,54)(6,57,62,55)(7,58,63,56)(8,59,64,53)(9,33,14,40)(10,34,15,37)(11,35,16,38)(12,36,13,39)(17,45,24,41)(18,46,21,42)(19,47,22,43)(20,48,23,44), (1,56,3,54)(2,57,4,59)(5,31,7,29)(6,51,8,49)(9,43,11,41)(10,46,12,48)(13,44,15,42)(14,47,16,45)(17,40,19,38)(18,36,20,34)(21,39,23,37)(22,35,24,33)(25,55,27,53)(26,60,28,58)(30,64,32,62)(50,61,52,63), (1,27,26,2)(3,25,28,4)(5,57,63,53)(6,56,64,60)(7,59,61,55)(8,54,62,58)(9,15,16,12)(10,11,13,14)(17,48,22,42)(18,41,23,47)(19,46,24,44)(20,43,21,45)(29,32,52,51)(30,50,49,31)(33,34,38,39)(35,36,40,37), (1,16)(2,10)(3,14)(4,12)(5,17)(6,23)(7,19)(8,21)(9,26)(11,28)(13,27)(15,25)(18,64)(20,62)(22,63)(24,61)(29,33)(30,39)(31,35)(32,37)(34,49)(36,51)(38,52)(40,50)(41,54)(42,59)(43,56)(44,57)(45,60)(46,53)(47,58)(48,55) );
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)], [(1,52,28,31),(2,49,25,32),(3,50,26,29),(4,51,27,30),(5,60,61,54),(6,57,62,55),(7,58,63,56),(8,59,64,53),(9,33,14,40),(10,34,15,37),(11,35,16,38),(12,36,13,39),(17,45,24,41),(18,46,21,42),(19,47,22,43),(20,48,23,44)], [(1,56,3,54),(2,57,4,59),(5,31,7,29),(6,51,8,49),(9,43,11,41),(10,46,12,48),(13,44,15,42),(14,47,16,45),(17,40,19,38),(18,36,20,34),(21,39,23,37),(22,35,24,33),(25,55,27,53),(26,60,28,58),(30,64,32,62),(50,61,52,63)], [(1,27,26,2),(3,25,28,4),(5,57,63,53),(6,56,64,60),(7,59,61,55),(8,54,62,58),(9,15,16,12),(10,11,13,14),(17,48,22,42),(18,41,23,47),(19,46,24,44),(20,43,21,45),(29,32,52,51),(30,50,49,31),(33,34,38,39),(35,36,40,37)], [(1,16),(2,10),(3,14),(4,12),(5,17),(6,23),(7,19),(8,21),(9,26),(11,28),(13,27),(15,25),(18,64),(20,62),(22,63),(24,61),(29,33),(30,39),(31,35),(32,37),(34,49),(36,51),(38,52),(40,50),(41,54),(42,59),(43,56),(44,57),(45,60),(46,53),(47,58),(48,55)]])
Matrix representation of C42.62C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 16 | 0 | 15 |
| 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 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 12 | 10 | 7 |
| 0 | 0 | 12 | 12 | 7 | 7 |
| 0 | 0 | 12 | 5 | 12 | 5 |
| 0 | 0 | 5 | 5 | 5 | 5 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 15 | 0 |
| 0 | 0 | 0 | 1 | 0 | 2 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 0 | 16 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,15,0,1,0,0,0,0,15,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,5,12,12,5,0,0,12,12,5,5,0,0,10,7,12,5,0,0,7,7,5,5],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,15,0,1,0,0,0,0,2,0,16],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,16,0,1,0,0,0,0,16,0,1,0,0,0,0,1,0,0,0,0,0,0,1] >;
C42.62C23 in GAP, Magma, Sage, TeX
C_4^2._{62}C_2^3 % in TeX
G:=Group("C4^2.62C2^3"); // GroupNames label
G:=SmallGroup(128,2080);
// by ID
G=gap.SmallGroup(128,2080);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,100,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=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=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations
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