p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.511C23, C4.322- 1+4, (C4×D8)⋊46C2, C8⋊4Q8⋊7C2, C4⋊C4.176D4, D4.Q8⋊48C2, Q8.Q8⋊46C2, D8⋊C4⋊28C2, D4⋊3Q8⋊13C2, D4⋊Q8⋊44C2, C2.64(D4○D8), C4⋊D8.12C2, (C2×Q8).136D4, D4.37(C4○D4), D4.2D4⋊46C2, C4⋊C8.135C22, C4⋊C4.263C23, (C2×C8).208C23, (C2×C4).562C24, (C4×C8).232C22, (C2×D8).92C22, C4⋊Q8.191C22, SD16⋊C4⋊45C2, C4.Q8.74C22, C8⋊C4.61C22, C2.70(Q8⋊5D4), (C4×D4).201C22, (C2×D4).273C23, (C4×Q8).193C22, (C2×Q8).257C23, C2.D8.206C22, C4⋊1D4.100C22, Q8⋊C4.88C22, (C2×SD16).69C22, C4.4D4.80C22, C22.822(C22×D4), C42.C2.65C22, D4⋊C4.211C22, C22.53C24⋊4C2, C2.101(D8⋊C22), C42.29C22⋊14C2, C42.28C22⋊21C2, C42.78C22⋊13C2, C4.263(C2×C4○D4), (C2×C4).638(C2×D4), SmallGroup(128,2102)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.511C23
G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=a2b2c, ece=bc, ede=b2d >
Subgroups: 360 in 179 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C4×D8, SD16⋊C4, D8⋊C4, C8⋊4Q8, C4⋊D8, D4.2D4, D4⋊Q8, D4.Q8, Q8.Q8, C42.78C22, C42.28C22, C42.29C22, D4⋊3Q8, C22.53C24, C42.511C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, D8⋊C22, D4○D8, C42.511C23
Character table of C42.511C23
| 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 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 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 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 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 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 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 |
| ρ27 | 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 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 63)(6 9 16 64)(7 10 13 61)(8 11 14 62)(29 36 37 41)(30 33 38 42)(31 34 39 43)(32 35 40 44)(45 51 56 60)(46 52 53 57)(47 49 54 58)(48 50 55 59)
(1 47)(2 48)(3 45)(4 46)(5 40)(6 37)(7 38)(8 39)(9 36)(10 33)(11 34)(12 35)(13 30)(14 31)(15 32)(16 29)(17 55)(18 56)(19 53)(20 54)(21 51)(22 52)(23 49)(24 50)(25 59)(26 60)(27 57)(28 58)(41 64)(42 61)(43 62)(44 63)
(1 56 18 47)(2 48 19 53)(3 54 20 45)(4 46 17 55)(5 35 13 42)(6 43 14 36)(7 33 15 44)(8 41 16 34)(9 31 62 37)(10 38 63 32)(11 29 64 39)(12 40 61 30)(21 49 28 60)(22 57 25 50)(23 51 26 58)(24 59 27 52)
(1 32)(2 29)(3 30)(4 31)(5 49)(6 50)(7 51)(8 52)(9 48)(10 45)(11 46)(12 47)(13 60)(14 57)(15 58)(16 59)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(53 62)(54 63)(55 64)(56 61)
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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,12,15,63)(6,9,16,64)(7,10,13,61)(8,11,14,62)(29,36,37,41)(30,33,38,42)(31,34,39,43)(32,35,40,44)(45,51,56,60)(46,52,53,57)(47,49,54,58)(48,50,55,59), (1,47)(2,48)(3,45)(4,46)(5,40)(6,37)(7,38)(8,39)(9,36)(10,33)(11,34)(12,35)(13,30)(14,31)(15,32)(16,29)(17,55)(18,56)(19,53)(20,54)(21,51)(22,52)(23,49)(24,50)(25,59)(26,60)(27,57)(28,58)(41,64)(42,61)(43,62)(44,63), (1,56,18,47)(2,48,19,53)(3,54,20,45)(4,46,17,55)(5,35,13,42)(6,43,14,36)(7,33,15,44)(8,41,16,34)(9,31,62,37)(10,38,63,32)(11,29,64,39)(12,40,61,30)(21,49,28,60)(22,57,25,50)(23,51,26,58)(24,59,27,52), (1,32)(2,29)(3,30)(4,31)(5,49)(6,50)(7,51)(8,52)(9,48)(10,45)(11,46)(12,47)(13,60)(14,57)(15,58)(16,59)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(53,62)(54,63)(55,64)(56,61)>;
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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,12,15,63)(6,9,16,64)(7,10,13,61)(8,11,14,62)(29,36,37,41)(30,33,38,42)(31,34,39,43)(32,35,40,44)(45,51,56,60)(46,52,53,57)(47,49,54,58)(48,50,55,59), (1,47)(2,48)(3,45)(4,46)(5,40)(6,37)(7,38)(8,39)(9,36)(10,33)(11,34)(12,35)(13,30)(14,31)(15,32)(16,29)(17,55)(18,56)(19,53)(20,54)(21,51)(22,52)(23,49)(24,50)(25,59)(26,60)(27,57)(28,58)(41,64)(42,61)(43,62)(44,63), (1,56,18,47)(2,48,19,53)(3,54,20,45)(4,46,17,55)(5,35,13,42)(6,43,14,36)(7,33,15,44)(8,41,16,34)(9,31,62,37)(10,38,63,32)(11,29,64,39)(12,40,61,30)(21,49,28,60)(22,57,25,50)(23,51,26,58)(24,59,27,52), (1,32)(2,29)(3,30)(4,31)(5,49)(6,50)(7,51)(8,52)(9,48)(10,45)(11,46)(12,47)(13,60)(14,57)(15,58)(16,59)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(25,41)(26,42)(27,43)(28,44)(53,62)(54,63)(55,64)(56,61) );
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)], [(1,28,20,23),(2,25,17,24),(3,26,18,21),(4,27,19,22),(5,12,15,63),(6,9,16,64),(7,10,13,61),(8,11,14,62),(29,36,37,41),(30,33,38,42),(31,34,39,43),(32,35,40,44),(45,51,56,60),(46,52,53,57),(47,49,54,58),(48,50,55,59)], [(1,47),(2,48),(3,45),(4,46),(5,40),(6,37),(7,38),(8,39),(9,36),(10,33),(11,34),(12,35),(13,30),(14,31),(15,32),(16,29),(17,55),(18,56),(19,53),(20,54),(21,51),(22,52),(23,49),(24,50),(25,59),(26,60),(27,57),(28,58),(41,64),(42,61),(43,62),(44,63)], [(1,56,18,47),(2,48,19,53),(3,54,20,45),(4,46,17,55),(5,35,13,42),(6,43,14,36),(7,33,15,44),(8,41,16,34),(9,31,62,37),(10,38,63,32),(11,29,64,39),(12,40,61,30),(21,49,28,60),(22,57,25,50),(23,51,26,58),(24,59,27,52)], [(1,32),(2,29),(3,30),(4,31),(5,49),(6,50),(7,51),(8,52),(9,48),(10,45),(11,46),(12,47),(13,60),(14,57),(15,58),(16,59),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(25,41),(26,42),(27,43),(28,44),(53,62),(54,63),(55,64),(56,61)]])
Matrix representation of C42.511C23 ►in GL6(𝔽17)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 16 |
| 0 | 0 | 0 | 16 | 1 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 6 | 11 |
| 0 | 0 | 14 | 6 | 0 | 11 |
| 0 | 0 | 14 | 3 | 3 | 8 |
| 0 | 0 | 0 | 3 | 14 | 14 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 6 | 11 |
| 0 | 0 | 14 | 11 | 6 | 0 |
| 0 | 0 | 11 | 14 | 9 | 14 |
| 0 | 0 | 14 | 14 | 3 | 3 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 16 | 0 |
G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,1,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,16,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,11,14,14,0,0,0,0,6,3,3,0,0,6,0,3,14,0,0,11,11,8,14],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,11,14,11,14,0,0,0,11,14,14,0,0,6,6,9,3,0,0,11,0,14,3],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,16,16,16,0,0,0,1,0,0] >;
C42.511C23 in GAP, Magma, Sage, TeX
C_4^2._{511}C_2^3 % in TeX
G:=Group("C4^2.511C2^3"); // GroupNames label
G:=SmallGroup(128,2102);
// by ID
G=gap.SmallGroup(128,2102);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,346,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e=b*c,e*d*e=b^2*d>;
// generators/relations
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