p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.509C23, C4.302- 1+4, C8⋊4Q8⋊5C2, C4⋊C4.174D4, Q8⋊9(C4○D4), Q8⋊3Q8⋊5C2, D4.Q8⋊47C2, C4⋊SD16⋊25C2, D4⋊Q8⋊43C2, (C4×SD16)⋊61C2, C4.4D8⋊36C2, (C2×Q8).242D4, Q8⋊6D4.9C2, C4⋊C4.436C23, C4⋊C8.133C22, C4.78(C8⋊C22), (C2×C4).560C24, (C2×C8).116C23, (C4×C8).296C22, C4⋊Q8.189C22, SD16⋊C4⋊44C2, C8⋊C4.59C22, C2.68(Q8⋊5D4), (C2×D4).271C23, (C4×D4).199C22, C4⋊1D4.99C22, (C2×Q8).404C23, (C4×Q8).191C22, C4.Q8.178C22, C2.D8.134C22, C2.100(D4○SD16), D4⋊C4.88C22, (C2×SD16).68C22, C22.820(C22×D4), C42.C2.64C22, Q8⋊C4.209C22, C42.29C22⋊13C2, C4.261(C2×C4○D4), (C2×C4).636(C2×D4), C2.88(C2×C8⋊C22), SmallGroup(128,2100)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.509C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2, e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >
Subgroups: 392 in 191 conjugacy classes, 88 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, D4⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C4×Q8, C4⋊D4, C42.C2, C42.C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×SD16, C2×C4○D4, C4×SD16, SD16⋊C4, C8⋊4Q8, C4⋊SD16, C4⋊SD16, D4⋊Q8, D4.Q8, C4.4D8, C42.29C22, Q8⋊6D4, Q8⋊3Q8, C42.509C23
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○SD16, C42.509C23
Character table of C42.509C23
| 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 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | -2 | -2 | -2 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -2 | 2 | -2 | -2 | -2 | -2 | 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 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2 | -2i | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | -2 | 2i | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 2i | 2 | -2i | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | -2i | 2 | 2i | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 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 | -4 | 0 | 4 | 0 | 0 | 0 | 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 | -4 | 0 | 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 |
| 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 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 55)(2 56)(3 53)(4 54)(5 38)(6 39)(7 40)(8 37)(9 36)(10 33)(11 34)(12 35)(13 32)(14 29)(15 30)(16 31)(17 49)(18 50)(19 51)(20 52)(21 45)(22 46)(23 47)(24 48)(25 59)(26 60)(27 57)(28 58)(41 63)(42 64)(43 61)(44 62)
(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 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 56 9 60)(6 53 10 57)(7 54 11 58)(8 55 12 59)(13 46 62 52)(14 47 63 49)(15 48 64 50)(16 45 61 51)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)
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), (1,55)(2,56)(3,53)(4,54)(5,38)(6,39)(7,40)(8,37)(9,36)(10,33)(11,34)(12,35)(13,32)(14,29)(15,30)(16,31)(17,49)(18,50)(19,51)(20,52)(21,45)(22,46)(23,47)(24,48)(25,59)(26,60)(27,57)(28,58)(41,63)(42,64)(43,61)(44,62), (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), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,56,9,60)(6,53,10,57)(7,54,11,58)(8,55,12,59)(13,46,62,52)(14,47,63,49)(15,48,64,50)(16,45,61,51)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40)>;
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), (1,55)(2,56)(3,53)(4,54)(5,38)(6,39)(7,40)(8,37)(9,36)(10,33)(11,34)(12,35)(13,32)(14,29)(15,30)(16,31)(17,49)(18,50)(19,51)(20,52)(21,45)(22,46)(23,47)(24,48)(25,59)(26,60)(27,57)(28,58)(41,63)(42,64)(43,61)(44,62), (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), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,56,9,60)(6,53,10,57)(7,54,11,58)(8,55,12,59)(13,46,62,52)(14,47,63,49)(15,48,64,50)(16,45,61,51)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40) );
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)], [(1,55),(2,56),(3,53),(4,54),(5,38),(6,39),(7,40),(8,37),(9,36),(10,33),(11,34),(12,35),(13,32),(14,29),(15,30),(16,31),(17,49),(18,50),(19,51),(20,52),(21,45),(22,46),(23,47),(24,48),(25,59),(26,60),(27,57),(28,58),(41,63),(42,64),(43,61),(44,62)], [(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)], [(1,41,25,29),(2,42,26,30),(3,43,27,31),(4,44,28,32),(5,56,9,60),(6,53,10,57),(7,54,11,58),(8,55,12,59),(13,46,62,52),(14,47,63,49),(15,48,64,50),(16,45,61,51),(17,35,23,37),(18,36,24,38),(19,33,21,39),(20,34,22,40)]])
Matrix representation of C42.509C23 ►in GL6(𝔽17)
| 16 | 2 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 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 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 7 | 0 |
| 0 | 0 | 13 | 0 | 0 | 10 |
| 0 | 0 | 10 | 0 | 0 | 13 |
| 0 | 0 | 0 | 7 | 13 | 0 |
| 13 | 8 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 7 | 0 |
| 0 | 0 | 4 | 0 | 0 | 7 |
| 0 | 0 | 7 | 0 | 0 | 4 |
| 0 | 0 | 0 | 7 | 13 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 14 | 1 | 1 |
| 0 | 0 | 14 | 14 | 1 | 16 |
| 0 | 0 | 16 | 16 | 3 | 14 |
| 0 | 0 | 16 | 1 | 14 | 14 |
G:=sub<GL(6,GF(17))| [16,16,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[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],[13,13,0,0,0,0,8,4,0,0,0,0,0,0,0,13,10,0,0,0,13,0,0,7,0,0,7,0,0,13,0,0,0,10,13,0],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,4,7,0,0,0,13,0,0,7,0,0,7,0,0,13,0,0,0,7,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,14,16,16,0,0,14,14,16,1,0,0,1,1,3,14,0,0,1,16,14,14] >;
C42.509C23 in GAP, Magma, Sage, TeX
C_4^2._{509}C_2^3 % in TeX
G:=Group("C4^2.509C2^3"); // GroupNames label
G:=SmallGroup(128,2100);
// by ID
G=gap.SmallGroup(128,2100);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,352,346,80,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2,e^2=b^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^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations
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