p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.480C23, C4.762+ 1+4, C8⋊6D4⋊18C2, C4⋊C4.376D4, D4⋊3Q8⋊5C2, D4⋊2Q8⋊20C2, C4⋊2Q16⋊41C2, (C4×SD16)⋊59C2, (C2×D4).326D4, D4⋊6D4.8C2, C8.D4⋊30C2, C22⋊C4.59D4, D4.32(C4○D4), D4.7D4⋊50C2, C4⋊C4.423C23, C4⋊C8.113C22, (C2×C4).523C24, (C2×C8).358C23, (C4×C8).294C22, C4.SD16⋊35C2, C23.340(C2×D4), C4⋊Q8.158C22, C4.Q8.63C22, C2.84(D4○SD16), (C2×D4).427C23, (C4×D4).172C22, C4.49(C8.C22), C22⋊C8.91C22, (C4×Q8).167C22, (C2×Q8).231C23, (C2×Q16).87C22, C2.159(D4⋊5D4), C22⋊Q8.94C22, D4⋊C4.77C22, C23.36D4⋊26C2, C23.47D4⋊19C2, (C22×C4).336C23, C22.783(C22×D4), Q8⋊C4.117C22, (C2×SD16).162C22, (C2×M4(2)).125C22, C4.248(C2×C4○D4), (C2×C4).616(C2×D4), C2.80(C2×C8.C22), (C2×C4⋊C4).675C22, (C2×C4○D4).221C22, SmallGroup(128,2063)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.480C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=a-1, dad=ab2, eae=a-1b2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2b2c, ede=b2d >
Subgroups: 376 in 194 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C2×C4○D4, C23.36D4, C8⋊6D4, C4×SD16, D4.7D4, C4⋊2Q16, C8.D4, D4⋊2Q8, C23.47D4, C4.SD16, D4⋊6D4, D4⋊3Q8, C42.480C23
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, C42.480C23
Character table of C42.480C23
| 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 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 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 | 0 | -2 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | 0 | 2 | -2 | -2 | 2 | 0 | 2 | -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 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | 2i | 0 | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | -2i | 0 | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 2 | 0 | -2 | 0 | 2 | -2 | 0 | 2i | 0 | 0 | 0 | -2i | 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 | 0 | 0 | -4 | 4 | 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 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 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 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 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 | 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
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 52 9 43)(2 49 10 44)(3 50 11 41)(4 51 12 42)(5 39 31 55)(6 40 32 56)(7 37 29 53)(8 38 30 54)(13 21 20 46)(14 22 17 47)(15 23 18 48)(16 24 19 45)(25 57 62 33)(26 58 63 34)(27 59 64 35)(28 60 61 36)
(1 37 3 39)(2 40 4 38)(5 52 7 50)(6 51 8 49)(9 53 11 55)(10 56 12 54)(13 36 15 34)(14 35 16 33)(17 59 19 57)(18 58 20 60)(21 61 23 63)(22 64 24 62)(25 47 27 45)(26 46 28 48)(29 41 31 43)(30 44 32 42)
(1 3)(2 12)(4 10)(5 37)(6 54)(7 39)(8 56)(9 11)(13 15)(14 19)(16 17)(18 20)(21 48)(22 24)(23 46)(25 35)(26 60)(27 33)(28 58)(29 55)(30 40)(31 53)(32 38)(34 61)(36 63)(41 52)(42 44)(43 50)(45 47)(49 51)(57 64)(59 62)
(1 16)(2 18)(3 14)(4 20)(5 25)(6 61)(7 27)(8 63)(9 19)(10 15)(11 17)(12 13)(21 42)(22 50)(23 44)(24 52)(26 30)(28 32)(29 64)(31 62)(33 55)(34 38)(35 53)(36 40)(37 59)(39 57)(41 47)(43 45)(46 51)(48 49)(54 58)(56 60)
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), (1,52,9,43)(2,49,10,44)(3,50,11,41)(4,51,12,42)(5,39,31,55)(6,40,32,56)(7,37,29,53)(8,38,30,54)(13,21,20,46)(14,22,17,47)(15,23,18,48)(16,24,19,45)(25,57,62,33)(26,58,63,34)(27,59,64,35)(28,60,61,36), (1,37,3,39)(2,40,4,38)(5,52,7,50)(6,51,8,49)(9,53,11,55)(10,56,12,54)(13,36,15,34)(14,35,16,33)(17,59,19,57)(18,58,20,60)(21,61,23,63)(22,64,24,62)(25,47,27,45)(26,46,28,48)(29,41,31,43)(30,44,32,42), (1,3)(2,12)(4,10)(5,37)(6,54)(7,39)(8,56)(9,11)(13,15)(14,19)(16,17)(18,20)(21,48)(22,24)(23,46)(25,35)(26,60)(27,33)(28,58)(29,55)(30,40)(31,53)(32,38)(34,61)(36,63)(41,52)(42,44)(43,50)(45,47)(49,51)(57,64)(59,62), (1,16)(2,18)(3,14)(4,20)(5,25)(6,61)(7,27)(8,63)(9,19)(10,15)(11,17)(12,13)(21,42)(22,50)(23,44)(24,52)(26,30)(28,32)(29,64)(31,62)(33,55)(34,38)(35,53)(36,40)(37,59)(39,57)(41,47)(43,45)(46,51)(48,49)(54,58)(56,60)>;
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), (1,52,9,43)(2,49,10,44)(3,50,11,41)(4,51,12,42)(5,39,31,55)(6,40,32,56)(7,37,29,53)(8,38,30,54)(13,21,20,46)(14,22,17,47)(15,23,18,48)(16,24,19,45)(25,57,62,33)(26,58,63,34)(27,59,64,35)(28,60,61,36), (1,37,3,39)(2,40,4,38)(5,52,7,50)(6,51,8,49)(9,53,11,55)(10,56,12,54)(13,36,15,34)(14,35,16,33)(17,59,19,57)(18,58,20,60)(21,61,23,63)(22,64,24,62)(25,47,27,45)(26,46,28,48)(29,41,31,43)(30,44,32,42), (1,3)(2,12)(4,10)(5,37)(6,54)(7,39)(8,56)(9,11)(13,15)(14,19)(16,17)(18,20)(21,48)(22,24)(23,46)(25,35)(26,60)(27,33)(28,58)(29,55)(30,40)(31,53)(32,38)(34,61)(36,63)(41,52)(42,44)(43,50)(45,47)(49,51)(57,64)(59,62), (1,16)(2,18)(3,14)(4,20)(5,25)(6,61)(7,27)(8,63)(9,19)(10,15)(11,17)(12,13)(21,42)(22,50)(23,44)(24,52)(26,30)(28,32)(29,64)(31,62)(33,55)(34,38)(35,53)(36,40)(37,59)(39,57)(41,47)(43,45)(46,51)(48,49)(54,58)(56,60) );
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)], [(1,52,9,43),(2,49,10,44),(3,50,11,41),(4,51,12,42),(5,39,31,55),(6,40,32,56),(7,37,29,53),(8,38,30,54),(13,21,20,46),(14,22,17,47),(15,23,18,48),(16,24,19,45),(25,57,62,33),(26,58,63,34),(27,59,64,35),(28,60,61,36)], [(1,37,3,39),(2,40,4,38),(5,52,7,50),(6,51,8,49),(9,53,11,55),(10,56,12,54),(13,36,15,34),(14,35,16,33),(17,59,19,57),(18,58,20,60),(21,61,23,63),(22,64,24,62),(25,47,27,45),(26,46,28,48),(29,41,31,43),(30,44,32,42)], [(1,3),(2,12),(4,10),(5,37),(6,54),(7,39),(8,56),(9,11),(13,15),(14,19),(16,17),(18,20),(21,48),(22,24),(23,46),(25,35),(26,60),(27,33),(28,58),(29,55),(30,40),(31,53),(32,38),(34,61),(36,63),(41,52),(42,44),(43,50),(45,47),(49,51),(57,64),(59,62)], [(1,16),(2,18),(3,14),(4,20),(5,25),(6,61),(7,27),(8,63),(9,19),(10,15),(11,17),(12,13),(21,42),(22,50),(23,44),(24,52),(26,30),(28,32),(29,64),(31,62),(33,55),(34,38),(35,53),(36,40),(37,59),(39,57),(41,47),(43,45),(46,51),(48,49),(54,58),(56,60)]])
Matrix representation of C42.480C23 ►in GL6(𝔽17)
| 1 | 6 | 0 | 0 | 0 | 0 |
| 11 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 10 |
| 0 | 0 | 1 | 0 | 7 | 0 |
| 0 | 0 | 0 | 10 | 0 | 1 |
| 0 | 0 | 7 | 0 | 16 | 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 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 1 | 5 | 12 |
| 0 | 0 | 1 | 1 | 12 | 12 |
| 0 | 0 | 5 | 12 | 1 | 16 |
| 0 | 0 | 12 | 12 | 16 | 16 |
| 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 11 | 16 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 16 |
| 0 | 0 | 10 | 0 | 1 | 0 |
| 0 | 0 | 0 | 16 | 0 | 10 |
| 0 | 0 | 1 | 0 | 7 | 0 |
G:=sub<GL(6,GF(17))| [1,11,0,0,0,0,6,16,0,0,0,0,0,0,0,1,0,7,0,0,16,0,10,0,0,0,0,7,0,16,0,0,10,0,1,0],[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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,16,1,5,12,0,0,1,1,12,12,0,0,5,12,1,16,0,0,12,12,16,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[11,1,0,0,0,0,16,6,0,0,0,0,0,0,0,10,0,1,0,0,7,0,16,0,0,0,0,1,0,7,0,0,16,0,10,0] >;
C42.480C23 in GAP, Magma, Sage, TeX
C_4^2._{480}C_2^3 % in TeX
G:=Group("C4^2.480C2^3"); // GroupNames label
G:=SmallGroup(128,2063);
// by ID
G=gap.SmallGroup(128,2063);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,2019,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,e*a*e=a^-1*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations
Export