p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.482C23, C4.782+ 1+4, C8⋊D4⋊49C2, C8⋊6D4⋊20C2, C4⋊C4.378D4, D4⋊3Q8⋊6C2, (C4×Q16)⋊44C2, C4.Q16⋊37C2, C4⋊SD16⋊24C2, (C2×D4).328D4, C2.53(D4○D8), C4.4D8⋊22C2, Q8⋊6D4.7C2, C22⋊C4.61D4, D4.7D4⋊51C2, C4⋊C8.115C22, C4⋊C4.425C23, (C2×C8).194C23, (C2×C4).525C24, (C4×C8).229C22, Q8.32(C4○D4), C23.342(C2×D4), C4⋊Q8.160C22, (C2×D4).247C23, (C4×D4).174C22, C4⋊D4.96C22, C4⋊1D4.93C22, C22.D8⋊30C2, C4.78(C8.C22), C22⋊C8.93C22, (C2×Q8).232C23, (C4×Q8).169C22, C2.161(D4⋊5D4), C2.D8.127C22, C22⋊Q8.95C22, D4⋊C4.15C22, C23.36D4⋊28C2, (C22×C4).338C23, (C2×Q16).137C22, Q8⋊C4.75C22, (C2×SD16).62C22, C22.785(C22×D4), (C2×M4(2)).127C22, C4.250(C2×C4○D4), (C2×C4).618(C2×D4), C2.81(C2×C8.C22), (C2×C4⋊C4).677C22, (C2×C4○D4).223C22, SmallGroup(128,2065)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.482C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2, d2=b2, ab=ba, cac-1=a-1, dad-1=ab2, eae=a-1b2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2b2c, ede=b2d >
Subgroups: 416 in 199 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, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, Q8⋊C4, C4⋊C8, C2.D8, C2.D8, C2×C4⋊C4, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊1D4, C4⋊1D4, C4⋊Q8, C2×M4(2), C2×SD16, C2×Q16, C2×C4○D4, C23.36D4, C8⋊6D4, C4×Q16, D4.7D4, C4⋊SD16, C8⋊D4, C4.Q16, C22.D8, C4.4D8, Q8⋊6D4, D4⋊3Q8, C42.482C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C8.C22, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, C2×C8.C22, D4○D8, C42.482C23
Character table of C42.482C23
| 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 | 2 | -2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | -2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | 2 | 0 | 0 | -2 | 0 | -2i | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | 2 | 0 | 0 | -2 | 0 | 2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 2i | -2 | 0 | 0 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | -2i | -2 | 0 | 0 | 2 | 0 | 2i | 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 | 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 |
| ρ26 | 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 | orthogonal lifted from 2+ 1+4 |
| 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 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ29 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
►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 52 9 43)(2 49 10 44)(3 50 11 41)(4 51 12 42)(5 39 31 55)(6 40 32 56)(7 37 29 53)(8 38 30 54)(13 21 20 46)(14 22 17 47)(15 23 18 48)(16 24 19 45)(25 57 62 33)(26 58 63 34)(27 59 64 35)(28 60 61 36)
(1 37 3 39)(2 40 4 38)(5 52 7 50)(6 51 8 49)(9 53 11 55)(10 56 12 54)(13 36 15 34)(14 35 16 33)(17 59 19 57)(18 58 20 60)(21 61 23 63)(22 64 24 62)(25 47 27 45)(26 46 28 48)(29 41 31 43)(30 44 32 42)
(1 20 9 13)(2 14 10 17)(3 18 11 15)(4 16 12 19)(5 58 31 34)(6 35 32 59)(7 60 29 36)(8 33 30 57)(21 43 46 52)(22 49 47 44)(23 41 48 50)(24 51 45 42)(25 38 62 54)(26 55 63 39)(27 40 64 56)(28 53 61 37)
(1 16)(2 18)(3 14)(4 20)(5 25)(6 61)(7 27)(8 63)(9 19)(10 15)(11 17)(12 13)(21 42)(22 50)(23 44)(24 52)(26 30)(28 32)(29 64)(31 62)(33 55)(34 38)(35 53)(36 40)(37 59)(39 57)(41 47)(43 45)(46 51)(48 49)(54 58)(56 60)
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), (1,52,9,43)(2,49,10,44)(3,50,11,41)(4,51,12,42)(5,39,31,55)(6,40,32,56)(7,37,29,53)(8,38,30,54)(13,21,20,46)(14,22,17,47)(15,23,18,48)(16,24,19,45)(25,57,62,33)(26,58,63,34)(27,59,64,35)(28,60,61,36), (1,37,3,39)(2,40,4,38)(5,52,7,50)(6,51,8,49)(9,53,11,55)(10,56,12,54)(13,36,15,34)(14,35,16,33)(17,59,19,57)(18,58,20,60)(21,61,23,63)(22,64,24,62)(25,47,27,45)(26,46,28,48)(29,41,31,43)(30,44,32,42), (1,20,9,13)(2,14,10,17)(3,18,11,15)(4,16,12,19)(5,58,31,34)(6,35,32,59)(7,60,29,36)(8,33,30,57)(21,43,46,52)(22,49,47,44)(23,41,48,50)(24,51,45,42)(25,38,62,54)(26,55,63,39)(27,40,64,56)(28,53,61,37), (1,16)(2,18)(3,14)(4,20)(5,25)(6,61)(7,27)(8,63)(9,19)(10,15)(11,17)(12,13)(21,42)(22,50)(23,44)(24,52)(26,30)(28,32)(29,64)(31,62)(33,55)(34,38)(35,53)(36,40)(37,59)(39,57)(41,47)(43,45)(46,51)(48,49)(54,58)(56,60)>;
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), (1,52,9,43)(2,49,10,44)(3,50,11,41)(4,51,12,42)(5,39,31,55)(6,40,32,56)(7,37,29,53)(8,38,30,54)(13,21,20,46)(14,22,17,47)(15,23,18,48)(16,24,19,45)(25,57,62,33)(26,58,63,34)(27,59,64,35)(28,60,61,36), (1,37,3,39)(2,40,4,38)(5,52,7,50)(6,51,8,49)(9,53,11,55)(10,56,12,54)(13,36,15,34)(14,35,16,33)(17,59,19,57)(18,58,20,60)(21,61,23,63)(22,64,24,62)(25,47,27,45)(26,46,28,48)(29,41,31,43)(30,44,32,42), (1,20,9,13)(2,14,10,17)(3,18,11,15)(4,16,12,19)(5,58,31,34)(6,35,32,59)(7,60,29,36)(8,33,30,57)(21,43,46,52)(22,49,47,44)(23,41,48,50)(24,51,45,42)(25,38,62,54)(26,55,63,39)(27,40,64,56)(28,53,61,37), (1,16)(2,18)(3,14)(4,20)(5,25)(6,61)(7,27)(8,63)(9,19)(10,15)(11,17)(12,13)(21,42)(22,50)(23,44)(24,52)(26,30)(28,32)(29,64)(31,62)(33,55)(34,38)(35,53)(36,40)(37,59)(39,57)(41,47)(43,45)(46,51)(48,49)(54,58)(56,60) );
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)], [(1,52,9,43),(2,49,10,44),(3,50,11,41),(4,51,12,42),(5,39,31,55),(6,40,32,56),(7,37,29,53),(8,38,30,54),(13,21,20,46),(14,22,17,47),(15,23,18,48),(16,24,19,45),(25,57,62,33),(26,58,63,34),(27,59,64,35),(28,60,61,36)], [(1,37,3,39),(2,40,4,38),(5,52,7,50),(6,51,8,49),(9,53,11,55),(10,56,12,54),(13,36,15,34),(14,35,16,33),(17,59,19,57),(18,58,20,60),(21,61,23,63),(22,64,24,62),(25,47,27,45),(26,46,28,48),(29,41,31,43),(30,44,32,42)], [(1,20,9,13),(2,14,10,17),(3,18,11,15),(4,16,12,19),(5,58,31,34),(6,35,32,59),(7,60,29,36),(8,33,30,57),(21,43,46,52),(22,49,47,44),(23,41,48,50),(24,51,45,42),(25,38,62,54),(26,55,63,39),(27,40,64,56),(28,53,61,37)], [(1,16),(2,18),(3,14),(4,20),(5,25),(6,61),(7,27),(8,63),(9,19),(10,15),(11,17),(12,13),(21,42),(22,50),(23,44),(24,52),(26,30),(28,32),(29,64),(31,62),(33,55),(34,38),(35,53),(36,40),(37,59),(39,57),(41,47),(43,45),(46,51),(48,49),(54,58),(56,60)]])
Matrix representation of C42.482C23 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 9 | 0 |
| 0 | 0 | 0 | 2 | 0 | 9 |
| 0 | 0 | 11 | 0 | 15 | 0 |
| 0 | 0 | 0 | 11 | 0 | 15 |
| 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 | 0 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 11 | 7 | 7 |
| 0 | 0 | 11 | 6 | 7 | 10 |
| 0 | 0 | 1 | 1 | 6 | 6 |
| 0 | 0 | 1 | 16 | 6 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 0 | 0 | 16 | 0 | 15 |
| 0 | 0 | 16 | 0 | 16 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 5 | 0 |
| 0 | 0 | 0 | 7 | 0 | 5 |
| 0 | 0 | 4 | 0 | 10 | 0 |
| 0 | 0 | 0 | 4 | 0 | 10 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,2,0,11,0,0,0,0,2,0,11,0,0,9,0,15,0,0,0,0,9,0,15],[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],[13,13,0,0,0,0,0,4,0,0,0,0,0,0,11,11,1,1,0,0,11,6,1,16,0,0,7,7,6,6,0,0,7,10,6,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,0,0,0,0,16,0,1,0,0,2,0,16,0,0,0,0,15,0,1],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,7,0,4,0,0,0,0,7,0,4,0,0,5,0,10,0,0,0,0,5,0,10] >;
C42.482C23 in GAP, Magma, Sage, TeX
C_4^2._{482}C_2^3 % in TeX
G:=Group("C4^2.482C2^3"); // GroupNames label
G:=SmallGroup(128,2065);
// by ID
G=gap.SmallGroup(128,2065);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,352,2019,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,e*a*e=a^-1*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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