p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.64C23, C4.852- 1+4, C8⋊Q8⋊30C2, C8⋊D4⋊53C2, C8⋊8D4⋊24C2, C8⋊9D4⋊29C2, C4⋊C4.382D4, Q8.Q8⋊43C2, D4⋊2Q8⋊22C2, (C2×D4).182D4, C8.37(C4○D4), C22⋊C4.65D4, C4⋊C4.255C23, C4⋊C8.123C22, (C2×C4).542C24, (C2×C8).109C23, C23.347(C2×D4), C4⋊Q8.174C22, SD16⋊C4⋊42C2, C2.95(D4⋊6D4), C8⋊C4.56C22, C4.Q8.70C22, C2.92(D4○SD16), (C4×D4).182C22, (C2×D4).259C23, (C4×Q8).181C22, (C2×Q8).244C23, M4(2)⋊C4⋊37C2, C2.D8.225C22, D4⋊C4.83C22, C23.25D4⋊33C2, C4⋊D4.108C22, C23.19D4⋊46C2, C23.47D4⋊22C2, C23.20D4⋊47C2, C22⋊C8.101C22, (C22×C8).293C22, Q8⋊C4.81C22, (C2×SD16).66C22, C22.802(C22×D4), C22⋊Q8.107C22, C42.C2.55C22, C2.97(D8⋊C22), (C22×C4).1170C23, C22.50C24⋊8C2, C42⋊C2.213C22, (C2×M4(2)).135C22, C22.47C24.4C2, C4.124(C2×C4○D4), (C2×C4).626(C2×D4), (C2×C4⋊C4).691C22, SmallGroup(128,2082)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.64C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=a2, ab=ba, cac-1=eae=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, ede=b2d >
Subgroups: 320 in 174 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C23.25D4, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, D4⋊2Q8, Q8.Q8, C23.19D4, C23.47D4, C23.20D4, C8⋊Q8, C22.47C24, C22.50C24, C42.64C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2- 1+4, D4⋊6D4, D8⋊C22, D4○SD16, C42.64C23
Character table of C42.64C23
| 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 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 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 | 2 | -2 | 0 | -2 | -2 | -2 | -2 | 0 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | -2 | -2 | 2 | 2 | 0 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | -2 | -2 | 2 | 2 | 0 | -2 | 0 | 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 | 2 | -2 | 0 | 0 | 2i | 0 | -2i | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | 2i | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | -2i | 0 | -2i | 2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 2i | 0 | 2i | -2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 | 0 | 0 | 0 | symplectic lifted from 2- 1+4, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 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 | 0 | 0 | 0 | 4i | -4i | 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 | 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
(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 20 39 23)(2 17 40 24)(3 18 37 21)(4 19 38 22)(5 30 61 28)(6 31 62 25)(7 32 63 26)(8 29 64 27)(9 54 15 59)(10 55 16 60)(11 56 13 57)(12 53 14 58)(33 43 49 46)(34 44 50 47)(35 41 51 48)(36 42 52 45)
(1 11 37 15)(2 16 38 12)(3 9 39 13)(4 14 40 10)(5 33 63 51)(6 52 64 34)(7 35 61 49)(8 50 62 36)(17 55 22 58)(18 59 23 56)(19 53 24 60)(20 57 21 54)(25 45 29 44)(26 41 30 46)(27 47 31 42)(28 43 32 48)
(1 50 3 52)(2 35 4 33)(5 58 7 60)(6 54 8 56)(9 29 11 31)(10 28 12 26)(13 25 15 27)(14 32 16 30)(17 48 19 46)(18 42 20 44)(21 45 23 47)(22 43 24 41)(34 37 36 39)(38 49 40 51)(53 63 55 61)(57 62 59 64)
(1 11)(2 16)(3 9)(4 14)(5 41)(6 47)(7 43)(8 45)(10 40)(12 38)(13 39)(15 37)(17 60)(18 54)(19 58)(20 56)(21 59)(22 53)(23 57)(24 55)(25 50)(26 33)(27 52)(28 35)(29 36)(30 51)(31 34)(32 49)(42 64)(44 62)(46 63)(48 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), (1,20,39,23)(2,17,40,24)(3,18,37,21)(4,19,38,22)(5,30,61,28)(6,31,62,25)(7,32,63,26)(8,29,64,27)(9,54,15,59)(10,55,16,60)(11,56,13,57)(12,53,14,58)(33,43,49,46)(34,44,50,47)(35,41,51,48)(36,42,52,45), (1,11,37,15)(2,16,38,12)(3,9,39,13)(4,14,40,10)(5,33,63,51)(6,52,64,34)(7,35,61,49)(8,50,62,36)(17,55,22,58)(18,59,23,56)(19,53,24,60)(20,57,21,54)(25,45,29,44)(26,41,30,46)(27,47,31,42)(28,43,32,48), (1,50,3,52)(2,35,4,33)(5,58,7,60)(6,54,8,56)(9,29,11,31)(10,28,12,26)(13,25,15,27)(14,32,16,30)(17,48,19,46)(18,42,20,44)(21,45,23,47)(22,43,24,41)(34,37,36,39)(38,49,40,51)(53,63,55,61)(57,62,59,64), (1,11)(2,16)(3,9)(4,14)(5,41)(6,47)(7,43)(8,45)(10,40)(12,38)(13,39)(15,37)(17,60)(18,54)(19,58)(20,56)(21,59)(22,53)(23,57)(24,55)(25,50)(26,33)(27,52)(28,35)(29,36)(30,51)(31,34)(32,49)(42,64)(44,62)(46,63)(48,61)>;
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), (1,20,39,23)(2,17,40,24)(3,18,37,21)(4,19,38,22)(5,30,61,28)(6,31,62,25)(7,32,63,26)(8,29,64,27)(9,54,15,59)(10,55,16,60)(11,56,13,57)(12,53,14,58)(33,43,49,46)(34,44,50,47)(35,41,51,48)(36,42,52,45), (1,11,37,15)(2,16,38,12)(3,9,39,13)(4,14,40,10)(5,33,63,51)(6,52,64,34)(7,35,61,49)(8,50,62,36)(17,55,22,58)(18,59,23,56)(19,53,24,60)(20,57,21,54)(25,45,29,44)(26,41,30,46)(27,47,31,42)(28,43,32,48), (1,50,3,52)(2,35,4,33)(5,58,7,60)(6,54,8,56)(9,29,11,31)(10,28,12,26)(13,25,15,27)(14,32,16,30)(17,48,19,46)(18,42,20,44)(21,45,23,47)(22,43,24,41)(34,37,36,39)(38,49,40,51)(53,63,55,61)(57,62,59,64), (1,11)(2,16)(3,9)(4,14)(5,41)(6,47)(7,43)(8,45)(10,40)(12,38)(13,39)(15,37)(17,60)(18,54)(19,58)(20,56)(21,59)(22,53)(23,57)(24,55)(25,50)(26,33)(27,52)(28,35)(29,36)(30,51)(31,34)(32,49)(42,64)(44,62)(46,63)(48,61) );
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)], [(1,20,39,23),(2,17,40,24),(3,18,37,21),(4,19,38,22),(5,30,61,28),(6,31,62,25),(7,32,63,26),(8,29,64,27),(9,54,15,59),(10,55,16,60),(11,56,13,57),(12,53,14,58),(33,43,49,46),(34,44,50,47),(35,41,51,48),(36,42,52,45)], [(1,11,37,15),(2,16,38,12),(3,9,39,13),(4,14,40,10),(5,33,63,51),(6,52,64,34),(7,35,61,49),(8,50,62,36),(17,55,22,58),(18,59,23,56),(19,53,24,60),(20,57,21,54),(25,45,29,44),(26,41,30,46),(27,47,31,42),(28,43,32,48)], [(1,50,3,52),(2,35,4,33),(5,58,7,60),(6,54,8,56),(9,29,11,31),(10,28,12,26),(13,25,15,27),(14,32,16,30),(17,48,19,46),(18,42,20,44),(21,45,23,47),(22,43,24,41),(34,37,36,39),(38,49,40,51),(53,63,55,61),(57,62,59,64)], [(1,11),(2,16),(3,9),(4,14),(5,41),(6,47),(7,43),(8,45),(10,40),(12,38),(13,39),(15,37),(17,60),(18,54),(19,58),(20,56),(21,59),(22,53),(23,57),(24,55),(25,50),(26,33),(27,52),(28,35),(29,36),(30,51),(31,34),(32,49),(42,64),(44,62),(46,63),(48,61)]])
Matrix representation of C42.64C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 4 | 4 |
| 0 | 0 | 0 | 12 | 15 | 13 |
| 0 | 0 | 4 | 4 | 5 | 0 |
| 0 | 0 | 15 | 13 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 15 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 1 | 1 |
| 0 | 0 | 14 | 14 | 8 | 0 |
| 0 | 0 | 0 | 16 | 14 | 11 |
| 0 | 0 | 9 | 1 | 0 | 3 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 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 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 1 | 1 |
| 0 | 0 | 0 | 3 | 8 | 16 |
| 0 | 0 | 1 | 1 | 14 | 0 |
| 0 | 0 | 8 | 16 | 0 | 14 |
G:=sub<GL(6,GF(17))| [4,0,0,0,0,0,0,13,0,0,0,0,0,0,12,0,4,15,0,0,0,12,4,13,0,0,4,15,5,0,0,0,4,13,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16],[0,2,0,0,0,0,8,0,0,0,0,0,0,0,3,14,0,9,0,0,0,14,16,1,0,0,1,8,14,0,0,0,1,0,11,3],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0],[0,2,0,0,0,0,9,0,0,0,0,0,0,0,3,0,1,8,0,0,0,3,1,16,0,0,1,8,14,0,0,0,1,16,0,14] >;
C42.64C23 in GAP, Magma, Sage, TeX
C_4^2._{64}C_2^3 % in TeX
G:=Group("C4^2.64C2^3"); // GroupNames label
G:=SmallGroup(128,2082);
// by ID
G=gap.SmallGroup(128,2082);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,436,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=a^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations
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