p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.423C23, C4.652- 1+4, C8⋊Q8⋊19C2, C4⋊C4.135D4, D4.Q8⋊25C2, C8.5Q8⋊5C2, D4⋊2Q8⋊11C2, D4⋊Q8⋊28C2, C2.32(D4○D8), C4.4D8⋊19C2, C4⋊C8.75C22, C22⋊C4.27D4, C4⋊C4.180C23, (C4×C8).117C22, (C2×C8).168C23, (C2×C4).439C24, C23.302(C2×D4), C4⋊Q8.123C22, C8⋊C4.32C22, C4.Q8.89C22, C2.49(D4○SD16), (C4×D4).121C22, (C2×D4).183C23, C4⋊D4.47C22, C4⋊1D4.70C22, C22.D8⋊24C2, C22⋊C8.66C22, C2.D8.109C22, D4⋊C4.53C22, C23.19D4⋊29C2, C23.46D4⋊13C2, (C22×C4).312C23, C22.699(C22×D4), C42.C2.26C22, C42.7C22⋊16C2, C23.41C23⋊8C2, C42.29C22⋊7C2, C42⋊C2.169C22, C22.34C24.3C2, C2.87(C23.38C23), (C2×C4).563(C2×D4), (C2×C4⋊C4).654C22, SmallGroup(128,1973)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.423C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2, ab=ba, cac=dad-1=a-1, eae=ab2, cbc=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >
Subgroups: 340 in 166 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, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42.C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C42.7C22, D4⋊Q8, D4⋊2Q8, D4.Q8, C22.D8, C23.46D4, C23.19D4, C4.4D8, C42.29C22, C8.5Q8, C8⋊Q8, C22.34C24, C23.41C23, C42.423C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2- 1+4, C23.38C23, D4○D8, D4○SD16, C42.423C23
Character table of C42.423C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 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 | 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 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 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 | orthogonal lifted from D4○D8 |
| ρ22 | 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 | orthogonal lifted from D4○D8 |
| ρ23 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ24 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ25 | 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 |
| 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 59 11 41)(2 60 12 42)(3 57 9 43)(4 58 10 44)(5 62 50 20)(6 63 51 17)(7 64 52 18)(8 61 49 19)(13 25 31 47)(14 26 32 48)(15 27 29 45)(16 28 30 46)(21 55 40 34)(22 56 37 35)(23 53 38 36)(24 54 39 33)
(1 4)(2 3)(5 19)(6 18)(7 17)(8 20)(9 12)(10 11)(13 14)(15 16)(21 54)(22 53)(23 56)(24 55)(25 48)(26 47)(27 46)(28 45)(29 30)(31 32)(33 40)(34 39)(35 38)(36 37)(41 58)(42 57)(43 60)(44 59)(49 62)(50 61)(51 64)(52 63)
(1 63 3 61)(2 62 4 64)(5 58 7 60)(6 57 8 59)(9 19 11 17)(10 18 12 20)(13 23 15 21)(14 22 16 24)(25 36 27 34)(26 35 28 33)(29 40 31 38)(30 39 32 37)(41 51 43 49)(42 50 44 52)(45 55 47 53)(46 54 48 56)
(1 25)(2 48)(3 27)(4 46)(5 22)(6 38)(7 24)(8 40)(9 45)(10 28)(11 47)(12 26)(13 41)(14 60)(15 43)(16 58)(17 53)(18 33)(19 55)(20 35)(21 49)(23 51)(29 57)(30 44)(31 59)(32 42)(34 61)(36 63)(37 50)(39 52)(54 64)(56 62)
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), (1,59,11,41)(2,60,12,42)(3,57,9,43)(4,58,10,44)(5,62,50,20)(6,63,51,17)(7,64,52,18)(8,61,49,19)(13,25,31,47)(14,26,32,48)(15,27,29,45)(16,28,30,46)(21,55,40,34)(22,56,37,35)(23,53,38,36)(24,54,39,33), (1,4)(2,3)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,14)(15,16)(21,54)(22,53)(23,56)(24,55)(25,48)(26,47)(27,46)(28,45)(29,30)(31,32)(33,40)(34,39)(35,38)(36,37)(41,58)(42,57)(43,60)(44,59)(49,62)(50,61)(51,64)(52,63), (1,63,3,61)(2,62,4,64)(5,58,7,60)(6,57,8,59)(9,19,11,17)(10,18,12,20)(13,23,15,21)(14,22,16,24)(25,36,27,34)(26,35,28,33)(29,40,31,38)(30,39,32,37)(41,51,43,49)(42,50,44,52)(45,55,47,53)(46,54,48,56), (1,25)(2,48)(3,27)(4,46)(5,22)(6,38)(7,24)(8,40)(9,45)(10,28)(11,47)(12,26)(13,41)(14,60)(15,43)(16,58)(17,53)(18,33)(19,55)(20,35)(21,49)(23,51)(29,57)(30,44)(31,59)(32,42)(34,61)(36,63)(37,50)(39,52)(54,64)(56,62)>;
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), (1,59,11,41)(2,60,12,42)(3,57,9,43)(4,58,10,44)(5,62,50,20)(6,63,51,17)(7,64,52,18)(8,61,49,19)(13,25,31,47)(14,26,32,48)(15,27,29,45)(16,28,30,46)(21,55,40,34)(22,56,37,35)(23,53,38,36)(24,54,39,33), (1,4)(2,3)(5,19)(6,18)(7,17)(8,20)(9,12)(10,11)(13,14)(15,16)(21,54)(22,53)(23,56)(24,55)(25,48)(26,47)(27,46)(28,45)(29,30)(31,32)(33,40)(34,39)(35,38)(36,37)(41,58)(42,57)(43,60)(44,59)(49,62)(50,61)(51,64)(52,63), (1,63,3,61)(2,62,4,64)(5,58,7,60)(6,57,8,59)(9,19,11,17)(10,18,12,20)(13,23,15,21)(14,22,16,24)(25,36,27,34)(26,35,28,33)(29,40,31,38)(30,39,32,37)(41,51,43,49)(42,50,44,52)(45,55,47,53)(46,54,48,56), (1,25)(2,48)(3,27)(4,46)(5,22)(6,38)(7,24)(8,40)(9,45)(10,28)(11,47)(12,26)(13,41)(14,60)(15,43)(16,58)(17,53)(18,33)(19,55)(20,35)(21,49)(23,51)(29,57)(30,44)(31,59)(32,42)(34,61)(36,63)(37,50)(39,52)(54,64)(56,62) );
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)], [(1,59,11,41),(2,60,12,42),(3,57,9,43),(4,58,10,44),(5,62,50,20),(6,63,51,17),(7,64,52,18),(8,61,49,19),(13,25,31,47),(14,26,32,48),(15,27,29,45),(16,28,30,46),(21,55,40,34),(22,56,37,35),(23,53,38,36),(24,54,39,33)], [(1,4),(2,3),(5,19),(6,18),(7,17),(8,20),(9,12),(10,11),(13,14),(15,16),(21,54),(22,53),(23,56),(24,55),(25,48),(26,47),(27,46),(28,45),(29,30),(31,32),(33,40),(34,39),(35,38),(36,37),(41,58),(42,57),(43,60),(44,59),(49,62),(50,61),(51,64),(52,63)], [(1,63,3,61),(2,62,4,64),(5,58,7,60),(6,57,8,59),(9,19,11,17),(10,18,12,20),(13,23,15,21),(14,22,16,24),(25,36,27,34),(26,35,28,33),(29,40,31,38),(30,39,32,37),(41,51,43,49),(42,50,44,52),(45,55,47,53),(46,54,48,56)], [(1,25),(2,48),(3,27),(4,46),(5,22),(6,38),(7,24),(8,40),(9,45),(10,28),(11,47),(12,26),(13,41),(14,60),(15,43),(16,58),(17,53),(18,33),(19,55),(20,35),(21,49),(23,51),(29,57),(30,44),(31,59),(32,42),(34,61),(36,63),(37,50),(39,52),(54,64),(56,62)]])
Matrix representation of C42.423C23 ►in GL8(𝔽17)
| 0 | 7 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 12 | 12 | 0 | 0 | 0 | 0 |
| 12 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 5 | 16 | 2 |
| 0 | 0 | 0 | 0 | 6 | 11 | 16 | 1 |
| 0 | 0 | 0 | 0 | 1 | 15 | 11 | 12 |
| 0 | 0 | 0 | 0 | 1 | 16 | 11 | 6 |
| 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 15 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 16 |
| 0 | 7 | 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 5 | 12 | 5 | 0 | 0 | 0 | 0 |
| 12 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 5 | 16 | 2 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 15 | 11 | 12 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 6 |
| 16 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 16 | 16 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 16 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 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 |
G:=sub<GL(8,GF(17))| [0,0,5,12,0,0,0,0,7,5,5,7,0,0,0,0,10,12,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,6,6,1,1,0,0,0,0,5,11,15,16,0,0,0,0,16,16,11,11,0,0,0,0,2,1,12,6],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,0,12,12,0,0,0,0,7,5,5,7,0,0,0,0,10,12,12,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,6,0,1,0,0,0,0,0,5,11,15,16,0,0,0,0,16,0,11,0,0,0,0,0,2,1,12,6],[16,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,15,1,1,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,7,0],[16,0,0,0,0,0,0,0,15,1,1,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.423C23 in GAP, Magma, Sage, TeX
C_4^2._{423}C_2^3 % in TeX
G:=Group("C4^2.423C2^3"); // GroupNames label
G:=SmallGroup(128,1973);
// by ID
G=gap.SmallGroup(128,1973);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,100,675,1018,4037,1027,124]);
// 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,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*c,d*e=e*d>;
// generators/relations
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