p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.507C23, C4.282- 1+4, (C4×D8)⋊45C2, C8⋊4Q8⋊3C2, D4⋊9(C4○D4), C4⋊D8⋊41C2, C4⋊C4.172D4, D4.Q8⋊46C2, D8⋊C4⋊27C2, D4⋊2Q8⋊25C2, D4⋊3Q8⋊12C2, Q8⋊6D4⋊11C2, C2.63(D4○D8), C4.4D8⋊23C2, (C2×Q8).240D4, C4⋊C8.131C22, C4⋊C4.434C23, C4.50(C8⋊C22), (C2×C4).558C24, (C4×C8).230C22, (C2×C8).114C23, (C2×D8).91C22, C4⋊Q8.187C22, C4.Q8.73C22, C8⋊C4.57C22, C2.66(Q8⋊5D4), (C2×D4).270C23, (C4×D4).197C22, C4⋊1D4.98C22, (C4×Q8).189C22, C2.D8.204C22, D4⋊C4.87C22, C22.818(C22×D4), C42.C2.63C22, C42.29C22⋊12C2, C4.259(C2×C4○D4), (C2×C4).634(C2×D4), C2.86(C2×C8⋊C22), SmallGroup(128,2098)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.507C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2c, ece=bc, ede=b2d >
Subgroups: 432 in 198 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, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C42.C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×D8, C2×C4○D4, C4×D8, D8⋊C4, C8⋊4Q8, C4⋊D8, C4⋊D8, D4⋊2Q8, D4.Q8, C4.4D8, C42.29C22, Q8⋊6D4, D4⋊3Q8, C42.507C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8⋊C22, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, C2×C8⋊C22, D4○D8, C42.507C23
Character table of C42.507C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 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 | 4 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 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 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ27 | 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 | orthogonal lifted from D4○D8 |
| 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 | orthogonal lifted from D4○D8 |
| ρ29 | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
►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 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)
(1 56)(2 53)(3 54)(4 55)(5 37)(6 38)(7 39)(8 40)(9 35)(10 36)(11 33)(12 34)(13 31)(14 32)(15 29)(16 30)(17 50)(18 51)(19 52)(20 49)(21 46)(22 47)(23 48)(24 45)(25 60)(26 57)(27 58)(28 59)(41 64)(42 61)(43 62)(44 63)
(1 53 3 55)(2 56 4 54)(5 44 7 42)(6 43 8 41)(9 32 11 30)(10 31 12 29)(13 36 15 34)(14 35 16 33)(17 45 19 47)(18 48 20 46)(21 49 23 51)(22 52 24 50)(25 57 27 59)(26 60 28 58)(37 61 39 63)(38 64 40 62)
(1 29)(2 30)(3 31)(4 32)(5 56)(6 53)(7 54)(8 55)(9 60)(10 57)(11 58)(12 59)(13 46)(14 47)(15 48)(16 45)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(49 63)(50 64)(51 61)(52 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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,56)(2,53)(3,54)(4,55)(5,37)(6,38)(7,39)(8,40)(9,35)(10,36)(11,33)(12,34)(13,31)(14,32)(15,29)(16,30)(17,50)(18,51)(19,52)(20,49)(21,46)(22,47)(23,48)(24,45)(25,60)(26,57)(27,58)(28,59)(41,64)(42,61)(43,62)(44,63), (1,53,3,55)(2,56,4,54)(5,44,7,42)(6,43,8,41)(9,32,11,30)(10,31,12,29)(13,36,15,34)(14,35,16,33)(17,45,19,47)(18,48,20,46)(21,49,23,51)(22,52,24,50)(25,57,27,59)(26,60,28,58)(37,61,39,63)(38,64,40,62), (1,29)(2,30)(3,31)(4,32)(5,56)(6,53)(7,54)(8,55)(9,60)(10,57)(11,58)(12,59)(13,46)(14,47)(15,48)(16,45)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(49,63)(50,64)(51,61)(52,62)>;
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), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,56)(2,53)(3,54)(4,55)(5,37)(6,38)(7,39)(8,40)(9,35)(10,36)(11,33)(12,34)(13,31)(14,32)(15,29)(16,30)(17,50)(18,51)(19,52)(20,49)(21,46)(22,47)(23,48)(24,45)(25,60)(26,57)(27,58)(28,59)(41,64)(42,61)(43,62)(44,63), (1,53,3,55)(2,56,4,54)(5,44,7,42)(6,43,8,41)(9,32,11,30)(10,31,12,29)(13,36,15,34)(14,35,16,33)(17,45,19,47)(18,48,20,46)(21,49,23,51)(22,52,24,50)(25,57,27,59)(26,60,28,58)(37,61,39,63)(38,64,40,62), (1,29)(2,30)(3,31)(4,32)(5,56)(6,53)(7,54)(8,55)(9,60)(10,57)(11,58)(12,59)(13,46)(14,47)(15,48)(16,45)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(49,63)(50,64)(51,61)(52,62) );
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)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(1,56),(2,53),(3,54),(4,55),(5,37),(6,38),(7,39),(8,40),(9,35),(10,36),(11,33),(12,34),(13,31),(14,32),(15,29),(16,30),(17,50),(18,51),(19,52),(20,49),(21,46),(22,47),(23,48),(24,45),(25,60),(26,57),(27,58),(28,59),(41,64),(42,61),(43,62),(44,63)], [(1,53,3,55),(2,56,4,54),(5,44,7,42),(6,43,8,41),(9,32,11,30),(10,31,12,29),(13,36,15,34),(14,35,16,33),(17,45,19,47),(18,48,20,46),(21,49,23,51),(22,52,24,50),(25,57,27,59),(26,60,28,58),(37,61,39,63),(38,64,40,62)], [(1,29),(2,30),(3,31),(4,32),(5,56),(6,53),(7,54),(8,55),(9,60),(10,57),(11,58),(12,59),(13,46),(14,47),(15,48),(16,45),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,41),(26,42),(27,43),(28,44),(49,63),(50,64),(51,61),(52,62)]])
Matrix representation of C42.507C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 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 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 13 | 4 |
| 0 | 0 | 0 | 4 | 13 | 13 |
| 0 | 0 | 4 | 4 | 0 | 13 |
| 0 | 0 | 13 | 4 | 13 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 4 | 13 |
| 0 | 0 | 0 | 4 | 13 | 13 |
| 0 | 0 | 13 | 4 | 13 | 0 |
| 0 | 0 | 4 | 4 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,13,0,4,13,0,0,0,4,4,4,0,0,13,13,0,13,0,0,4,13,13,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,13,4,0,0,0,4,4,4,0,0,4,13,13,0,0,0,13,13,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
C42.507C23 in GAP, Magma, Sage, TeX
C_4^2._{507}C_2^3 % in TeX
G:=Group("C4^2.507C2^3"); // GroupNames label
G:=SmallGroup(128,2098);
// by ID
G=gap.SmallGroup(128,2098);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,346,80,4037,1027,124]);
// 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,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e=b*c,e*d*e=b^2*d>;
// generators/relations
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