p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.512C23, C4.332- 1+4, C8⋊4Q8⋊8C2, C4⋊C4.177D4, Q8⋊3Q8⋊6C2, D4.Q8⋊49C2, Q8.Q8⋊47C2, D4⋊2Q8⋊26C2, (C4×SD16)⋊62C2, (C2×Q8).137D4, C4⋊SD16.1C2, Q16⋊C4⋊28C2, C4⋊C4.264C23, C4⋊C8.136C22, (C2×C8).371C23, (C2×C4).563C24, (C4×C8).298C22, Q8.37(C4○D4), Q8.D4⋊46C2, C4⋊Q8.192C22, SD16⋊C4⋊46C2, C8⋊C4.62C22, C2.71(Q8⋊5D4), (C4×D4).202C22, (C2×D4).274C23, (C4×Q8).194C22, (C2×Q16).91C22, (C2×Q8).405C23, C4.Q8.180C22, C2.D8.135C22, C2.102(D4○SD16), C4⋊1D4.101C22, Q8⋊C4.89C22, (C2×SD16).70C22, C4.4D4.81C22, C22.823(C22×D4), C42.C2.66C22, D4⋊C4.212C22, C2.102(D8⋊C22), C42.78C22⋊23C2, C42.28C22⋊22C2, C22.53C24.4C2, C42.29C22.1C2, C4.264(C2×C4○D4), (C2×C4).639(C2×D4), SmallGroup(128,2103)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.512C23
G = < a,b,c,d,e | a4=b4=c2=1, d2=a2b2, e2=b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2b2c, ece-1=bc, ede-1=b2d >
Subgroups: 320 in 172 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C4×D4, C4×Q8, C4×Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C4⋊1D4, C4⋊Q8, C4⋊Q8, C2×SD16, C2×Q16, C4×SD16, SD16⋊C4, Q16⋊C4, C8⋊4Q8, C4⋊SD16, Q8.D4, D4⋊2Q8, D4.Q8, Q8.Q8, C42.78C22, C42.28C22, C42.29C22, Q8⋊3Q8, C22.53C24, C42.512C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊5D4, D8⋊C22, D4○SD16, C42.512C23
Character table of C42.512C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 0 | -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 | 2 | -2 | 2 | -2 | -2 | 0 | -2 | 0 | 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 | -2 | -2 | -2 | -2 | 2 | 0 | 2 | 0 | -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 | -2 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | 2 | 0 | 0 | -2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | 2 | 0 | 0 | 2i | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | -2i | 0 | -2 | 0 | 0 | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | 0 | -2 | 0 | 2i | 0 | -2 | 0 | 0 | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 2i | 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 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | -4i | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 4i | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ28 | 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 | 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 | 2√-2 | 0 | -2√-2 | 0 | 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 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)
(5 10)(6 11)(7 12)(8 9)(13 63)(14 64)(15 61)(16 62)(21 26)(22 27)(23 28)(24 25)(29 36)(30 33)(31 34)(32 35)(37 41)(38 42)(39 43)(40 44)(45 54)(46 55)(47 56)(48 53)(49 51)(50 52)(57 59)(58 60)
(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 40 20 32)(2 37 17 29)(3 38 18 30)(4 39 19 31)(5 49 15 58)(6 50 16 59)(7 51 13 60)(8 52 14 57)(9 48 64 55)(10 45 61 56)(11 46 62 53)(12 47 63 54)(21 42 26 33)(22 43 27 34)(23 44 28 35)(24 41 25 36)
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), (5,10)(6,11)(7,12)(8,9)(13,63)(14,64)(15,61)(16,62)(21,26)(22,27)(23,28)(24,25)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(45,54)(46,55)(47,56)(48,53)(49,51)(50,52)(57,59)(58,60), (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), (1,40,20,32)(2,37,17,29)(3,38,18,30)(4,39,19,31)(5,49,15,58)(6,50,16,59)(7,51,13,60)(8,52,14,57)(9,48,64,55)(10,45,61,56)(11,46,62,53)(12,47,63,54)(21,42,26,33)(22,43,27,34)(23,44,28,35)(24,41,25,36)>;
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), (5,10)(6,11)(7,12)(8,9)(13,63)(14,64)(15,61)(16,62)(21,26)(22,27)(23,28)(24,25)(29,36)(30,33)(31,34)(32,35)(37,41)(38,42)(39,43)(40,44)(45,54)(46,55)(47,56)(48,53)(49,51)(50,52)(57,59)(58,60), (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), (1,40,20,32)(2,37,17,29)(3,38,18,30)(4,39,19,31)(5,49,15,58)(6,50,16,59)(7,51,13,60)(8,52,14,57)(9,48,64,55)(10,45,61,56)(11,46,62,53)(12,47,63,54)(21,42,26,33)(22,43,27,34)(23,44,28,35)(24,41,25,36) );
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)], [(5,10),(6,11),(7,12),(8,9),(13,63),(14,64),(15,61),(16,62),(21,26),(22,27),(23,28),(24,25),(29,36),(30,33),(31,34),(32,35),(37,41),(38,42),(39,43),(40,44),(45,54),(46,55),(47,56),(48,53),(49,51),(50,52),(57,59),(58,60)], [(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)], [(1,40,20,32),(2,37,17,29),(3,38,18,30),(4,39,19,31),(5,49,15,58),(6,50,16,59),(7,51,13,60),(8,52,14,57),(9,48,64,55),(10,45,61,56),(11,46,62,53),(12,47,63,54),(21,42,26,33),(22,43,27,34),(23,44,28,35),(24,41,25,36)]])
Matrix representation of C42.512C23 ►in GL6(𝔽17)
| 13 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 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 | 4 | 13 | 0 | 16 |
| 0 | 0 | 0 | 13 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 16 | 0 | 0 | 0 | 0 |
| 14 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 3 | 11 | 0 | 7 |
| 0 | 0 | 0 | 5 | 3 | 14 |
| 0 | 0 | 5 | 12 | 14 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 15 | 0 |
| 0 | 0 | 0 | 0 | 16 | 1 |
| 0 | 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 16 | 13 | 0 |
G:=sub<GL(6,GF(17))| [13,13,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,4,0,0,0,15,16,13,13,0,0,0,0,0,1,0,0,0,0,16,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,14,0,0,0,0,16,8,0,0,0,0,0,0,0,3,0,5,0,0,0,11,5,12,0,0,7,0,3,14,0,0,7,7,14,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,15,16,13,13,0,0,0,1,0,0] >;
C42.512C23 in GAP, Magma, Sage, TeX
C_4^2._{512}C_2^3 % in TeX
G:=Group("C4^2.512C2^3"); // GroupNames label
G:=SmallGroup(128,2103);
// by ID
G=gap.SmallGroup(128,2103);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,352,346,304,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=a^2*b^2,e^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^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations
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