p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.426C23, C4.682- 1+4, C8⋊Q8⋊22C2, C4⋊C4.138D4, C8⋊3Q8⋊24C2, D4.Q8⋊27C2, Q8.Q8⋊27C2, Q8⋊Q8⋊12C2, D4⋊2Q8⋊12C2, C4⋊C8.78C22, C22⋊C4.30D4, C4⋊C4.183C23, (C2×C8).337C23, (C4×C8).270C22, (C2×C4).442C24, C23.305(C2×D4), C4⋊Q8.126C22, C8⋊C4.35C22, C4.Q8.90C22, C2.51(D4○SD16), (C4×D4).123C22, (C2×D4).185C23, C4⋊D4.49C22, C22⋊C8.69C22, (C4×Q8).120C22, (C2×Q8).173C23, C2.D8.110C22, C22⋊Q8.49C22, D4⋊C4.55C22, C23.47D4⋊13C2, C23.20D4⋊30C2, (C22×C4).315C23, C4.4D4.44C22, C23.19D4.5C2, C23.46D4.3C2, C22.702(C22×D4), C42.C2.29C22, C42.7C22⋊18C2, Q8⋊C4.107C22, C42.28C22⋊11C2, C23.41C23⋊10C2, C42.78C22⋊16C2, C42⋊C2.172C22, C22.36C24.4C2, C2.90(C23.38C23), (C2×C4).566(C2×D4), (C2×C4⋊C4).657C22, SmallGroup(128,1976)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.426C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2, ab=ba, cac=dad-1=a-1b2, eae=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, de=ed >
Subgroups: 300 in 160 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C4⋊Q8, C42.7C22, Q8⋊Q8, D4⋊2Q8, D4.Q8, Q8.Q8, C23.46D4, C23.19D4, C23.47D4, C23.20D4, C42.78C22, C42.28C22, C8⋊3Q8, C8⋊Q8, C22.36C24, C23.41C23, C42.426C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2- 1+4, C23.38C23, D4○SD16, C42.426C23
Character table of C42.426C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | -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 | -2 | -2 | 2 | 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 | -2 | -2 | -2 | -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 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | -4 | 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 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 4 | 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 |
| ρ23 | 4 | -4 | -4 | 4 | 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 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ25 | 4 | -4 | -4 | 4 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√-2 | 0 | -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 53 47 58)(2 54 48 59)(3 55 45 60)(4 56 46 57)(5 42 61 52)(6 43 62 49)(7 44 63 50)(8 41 64 51)(9 34 21 37)(10 35 22 38)(11 36 23 39)(12 33 24 40)(13 28 18 31)(14 25 19 32)(15 26 20 29)(16 27 17 30)
(2 46)(4 48)(5 52)(6 41)(7 50)(8 43)(9 39)(10 35)(11 37)(12 33)(13 20)(15 18)(21 36)(22 38)(23 34)(24 40)(25 32)(26 28)(27 30)(29 31)(42 61)(44 63)(49 64)(51 62)(53 58)(54 56)(55 60)(57 59)
(1 50 3 52)(2 43 4 41)(5 58 7 60)(6 56 8 54)(9 18 11 20)(10 16 12 14)(13 23 15 21)(17 24 19 22)(25 38 27 40)(26 34 28 36)(29 37 31 39)(30 33 32 35)(42 47 44 45)(46 51 48 49)(53 63 55 61)(57 64 59 62)
(1 13)(2 19)(3 15)(4 17)(5 37)(6 35)(7 39)(8 33)(9 42)(10 49)(11 44)(12 51)(14 48)(16 46)(18 47)(20 45)(21 52)(22 43)(23 50)(24 41)(25 59)(26 55)(27 57)(28 53)(29 60)(30 56)(31 58)(32 54)(34 61)(36 63)(38 62)(40 64)
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), (1,53,47,58)(2,54,48,59)(3,55,45,60)(4,56,46,57)(5,42,61,52)(6,43,62,49)(7,44,63,50)(8,41,64,51)(9,34,21,37)(10,35,22,38)(11,36,23,39)(12,33,24,40)(13,28,18,31)(14,25,19,32)(15,26,20,29)(16,27,17,30), (2,46)(4,48)(5,52)(6,41)(7,50)(8,43)(9,39)(10,35)(11,37)(12,33)(13,20)(15,18)(21,36)(22,38)(23,34)(24,40)(25,32)(26,28)(27,30)(29,31)(42,61)(44,63)(49,64)(51,62)(53,58)(54,56)(55,60)(57,59), (1,50,3,52)(2,43,4,41)(5,58,7,60)(6,56,8,54)(9,18,11,20)(10,16,12,14)(13,23,15,21)(17,24,19,22)(25,38,27,40)(26,34,28,36)(29,37,31,39)(30,33,32,35)(42,47,44,45)(46,51,48,49)(53,63,55,61)(57,64,59,62), (1,13)(2,19)(3,15)(4,17)(5,37)(6,35)(7,39)(8,33)(9,42)(10,49)(11,44)(12,51)(14,48)(16,46)(18,47)(20,45)(21,52)(22,43)(23,50)(24,41)(25,59)(26,55)(27,57)(28,53)(29,60)(30,56)(31,58)(32,54)(34,61)(36,63)(38,62)(40,64)>;
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), (1,53,47,58)(2,54,48,59)(3,55,45,60)(4,56,46,57)(5,42,61,52)(6,43,62,49)(7,44,63,50)(8,41,64,51)(9,34,21,37)(10,35,22,38)(11,36,23,39)(12,33,24,40)(13,28,18,31)(14,25,19,32)(15,26,20,29)(16,27,17,30), (2,46)(4,48)(5,52)(6,41)(7,50)(8,43)(9,39)(10,35)(11,37)(12,33)(13,20)(15,18)(21,36)(22,38)(23,34)(24,40)(25,32)(26,28)(27,30)(29,31)(42,61)(44,63)(49,64)(51,62)(53,58)(54,56)(55,60)(57,59), (1,50,3,52)(2,43,4,41)(5,58,7,60)(6,56,8,54)(9,18,11,20)(10,16,12,14)(13,23,15,21)(17,24,19,22)(25,38,27,40)(26,34,28,36)(29,37,31,39)(30,33,32,35)(42,47,44,45)(46,51,48,49)(53,63,55,61)(57,64,59,62), (1,13)(2,19)(3,15)(4,17)(5,37)(6,35)(7,39)(8,33)(9,42)(10,49)(11,44)(12,51)(14,48)(16,46)(18,47)(20,45)(21,52)(22,43)(23,50)(24,41)(25,59)(26,55)(27,57)(28,53)(29,60)(30,56)(31,58)(32,54)(34,61)(36,63)(38,62)(40,64) );
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)], [(1,53,47,58),(2,54,48,59),(3,55,45,60),(4,56,46,57),(5,42,61,52),(6,43,62,49),(7,44,63,50),(8,41,64,51),(9,34,21,37),(10,35,22,38),(11,36,23,39),(12,33,24,40),(13,28,18,31),(14,25,19,32),(15,26,20,29),(16,27,17,30)], [(2,46),(4,48),(5,52),(6,41),(7,50),(8,43),(9,39),(10,35),(11,37),(12,33),(13,20),(15,18),(21,36),(22,38),(23,34),(24,40),(25,32),(26,28),(27,30),(29,31),(42,61),(44,63),(49,64),(51,62),(53,58),(54,56),(55,60),(57,59)], [(1,50,3,52),(2,43,4,41),(5,58,7,60),(6,56,8,54),(9,18,11,20),(10,16,12,14),(13,23,15,21),(17,24,19,22),(25,38,27,40),(26,34,28,36),(29,37,31,39),(30,33,32,35),(42,47,44,45),(46,51,48,49),(53,63,55,61),(57,64,59,62)], [(1,13),(2,19),(3,15),(4,17),(5,37),(6,35),(7,39),(8,33),(9,42),(10,49),(11,44),(12,51),(14,48),(16,46),(18,47),(20,45),(21,52),(22,43),(23,50),(24,41),(25,59),(26,55),(27,57),(28,53),(29,60),(30,56),(31,58),(32,54),(34,61),(36,63),(38,62),(40,64)]])
Matrix representation of C42.426C23 ►in GL8(𝔽17)
| 0 | 10 | 16 | 0 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 1 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 11 | 6 | 0 |
| 0 | 0 | 0 | 0 | 11 | 3 | 0 | 6 |
| 0 | 0 | 0 | 0 | 15 | 6 | 14 | 6 |
| 0 | 0 | 0 | 0 | 6 | 15 | 6 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
| 1 | 1 | 5 | 12 | 0 | 0 | 0 | 0 |
| 1 | 16 | 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 12 | 1 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 9 | 13 | 0 |
| 0 | 0 | 0 | 0 | 9 | 15 | 0 | 13 |
| 0 | 0 | 0 | 0 | 13 | 8 | 2 | 8 |
| 0 | 0 | 0 | 0 | 8 | 13 | 8 | 2 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(17))| [0,7,1,0,0,0,0,0,10,0,0,1,0,0,0,0,16,0,0,10,0,0,0,0,0,16,7,0,0,0,0,0,0,0,0,0,3,11,15,6,0,0,0,0,11,3,6,15,0,0,0,0,6,0,14,6,0,0,0,0,0,6,6,14],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1],[1,1,5,12,0,0,0,0,1,16,12,12,0,0,0,0,5,12,1,1,0,0,0,0,12,12,1,16,0,0,0,0,0,0,0,0,15,9,13,8,0,0,0,0,9,15,8,13,0,0,0,0,13,0,2,8,0,0,0,0,0,13,8,2],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C42.426C23 in GAP, Magma, Sage, TeX
C_4^2._{426}C_2^3 % in TeX
G:=Group("C4^2.426C2^3"); // GroupNames label
G:=SmallGroup(128,1976);
// by ID
G=gap.SmallGroup(128,1976);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,436,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1*b^2,e*a*e=a*b^2,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*b^2*c,d*e=e*d>;
// generators/relations
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