p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.476C23, C4.722+ 1+4, C8⋊D4⋊46C2, C8⋊6D4⋊15C2, C4⋊C4.164D4, (C4×Q16)⋊42C2, Q8.Q8⋊39C2, (C2×D4).178D4, C8.D4⋊28C2, Q8⋊5D4.5C2, C2.51(Q8○D8), D4.7D4⋊49C2, C4⋊C4.419C23, C4⋊C8.109C22, (C2×C4).519C24, (C4×C8).226C22, (C2×C8).105C23, Q8.29(C4○D4), Q8.D4⋊45C2, C22⋊Q16⋊33C2, C22⋊C4.174D4, C23.336(C2×D4), C4.Q8.61C22, (C2×D4).243C23, (C4×D4).168C22, C4⋊D4.92C22, C22⋊C8.87C22, (C4×Q8).164C22, (C2×Q8).228C23, C2.155(D4⋊5D4), C2.D8.124C22, C22⋊Q8.90C22, D4⋊C4.14C22, C23.36D4⋊24C2, C23.48D4⋊29C2, C23.19D4⋊41C2, C23.38D4⋊17C2, (C22×C4).332C23, (C2×Q16).135C22, Q8⋊C4.14C22, (C2×SD16).60C22, C4.4D4.73C22, C22.779(C22×D4), C42.C2.44C22, C2.93(D8⋊C22), C22.46C24⋊7C2, (C22×Q8).348C22, C42⋊C2.197C22, C42.78C22⋊12C2, (C2×M4(2)).121C22, C4.244(C2×C4○D4), (C2×C4).930(C2×D4), (C2×C4⋊C4).673C22, (C2×C4○D4).219C22, SmallGroup(128,2059)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.476C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=b2, ab=ba, cac-1=a-1b2, dad-1=ab2, eae=a-1, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >
Subgroups: 360 in 190 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C2×M4(2), C2×SD16, C2×Q16, C22×Q8, C2×C4○D4, C23.36D4, C23.38D4, C8⋊6D4, C4×Q16, C22⋊Q16, D4.7D4, Q8.D4, C8⋊D4, C8.D4, Q8.Q8, C23.19D4, C23.48D4, C42.78C22, Q8⋊5D4, C22.46C24, C42.476C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, Q8○D8, C42.476C23
Character table of C42.476C23
| 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 | 0 | 2 | 2 | 0 | 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 | 0 | 2 | -2 | 0 | -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 | 0 | -2 | 2 | 0 | -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 | 0 | -2 | -2 | 0 | 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 | -2 | 0 | -2i | -2 | 0 | 0 | 2 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | -2i | 2 | 0 | 0 | -2 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | 2 | 0 | 0 | -2 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 2i | -2 | 0 | 0 | 2 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ26 | 4 | 4 | -4 | -4 | 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 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| 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 | 2√2 | -2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 4i | 0 | 0 | -4i | 0 | 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 | -4i | 0 | 0 | 4i | 0 | 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
(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 45 17 41)(2 46 18 42)(3 47 19 43)(4 48 20 44)(5 38 34 62)(6 39 35 63)(7 40 36 64)(8 37 33 61)(9 52 22 14)(10 49 23 15)(11 50 24 16)(12 51 21 13)(25 54 58 31)(26 55 59 32)(27 56 60 29)(28 53 57 30)
(1 32 19 53)(2 54 20 29)(3 30 17 55)(4 56 18 31)(5 52 36 16)(6 13 33 49)(7 50 34 14)(8 15 35 51)(9 64 24 38)(10 39 21 61)(11 62 22 40)(12 37 23 63)(25 44 60 46)(26 47 57 41)(27 42 58 48)(28 45 59 43)
(1 11 17 24)(2 21 18 12)(3 9 19 22)(4 23 20 10)(5 32 34 55)(6 56 35 29)(7 30 36 53)(8 54 33 31)(13 46 51 42)(14 43 52 47)(15 48 49 44)(16 41 50 45)(25 61 58 37)(26 38 59 62)(27 63 60 39)(28 40 57 64)
(1 24)(2 23)(3 22)(4 21)(5 57)(6 60)(7 59)(8 58)(9 19)(10 18)(11 17)(12 20)(13 48)(14 47)(15 46)(16 45)(25 33)(26 36)(27 35)(28 34)(29 39)(30 38)(31 37)(32 40)(41 50)(42 49)(43 52)(44 51)(53 62)(54 61)(55 64)(56 63)
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), (1,45,17,41)(2,46,18,42)(3,47,19,43)(4,48,20,44)(5,38,34,62)(6,39,35,63)(7,40,36,64)(8,37,33,61)(9,52,22,14)(10,49,23,15)(11,50,24,16)(12,51,21,13)(25,54,58,31)(26,55,59,32)(27,56,60,29)(28,53,57,30), (1,32,19,53)(2,54,20,29)(3,30,17,55)(4,56,18,31)(5,52,36,16)(6,13,33,49)(7,50,34,14)(8,15,35,51)(9,64,24,38)(10,39,21,61)(11,62,22,40)(12,37,23,63)(25,44,60,46)(26,47,57,41)(27,42,58,48)(28,45,59,43), (1,11,17,24)(2,21,18,12)(3,9,19,22)(4,23,20,10)(5,32,34,55)(6,56,35,29)(7,30,36,53)(8,54,33,31)(13,46,51,42)(14,43,52,47)(15,48,49,44)(16,41,50,45)(25,61,58,37)(26,38,59,62)(27,63,60,39)(28,40,57,64), (1,24)(2,23)(3,22)(4,21)(5,57)(6,60)(7,59)(8,58)(9,19)(10,18)(11,17)(12,20)(13,48)(14,47)(15,46)(16,45)(25,33)(26,36)(27,35)(28,34)(29,39)(30,38)(31,37)(32,40)(41,50)(42,49)(43,52)(44,51)(53,62)(54,61)(55,64)(56,63)>;
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), (1,45,17,41)(2,46,18,42)(3,47,19,43)(4,48,20,44)(5,38,34,62)(6,39,35,63)(7,40,36,64)(8,37,33,61)(9,52,22,14)(10,49,23,15)(11,50,24,16)(12,51,21,13)(25,54,58,31)(26,55,59,32)(27,56,60,29)(28,53,57,30), (1,32,19,53)(2,54,20,29)(3,30,17,55)(4,56,18,31)(5,52,36,16)(6,13,33,49)(7,50,34,14)(8,15,35,51)(9,64,24,38)(10,39,21,61)(11,62,22,40)(12,37,23,63)(25,44,60,46)(26,47,57,41)(27,42,58,48)(28,45,59,43), (1,11,17,24)(2,21,18,12)(3,9,19,22)(4,23,20,10)(5,32,34,55)(6,56,35,29)(7,30,36,53)(8,54,33,31)(13,46,51,42)(14,43,52,47)(15,48,49,44)(16,41,50,45)(25,61,58,37)(26,38,59,62)(27,63,60,39)(28,40,57,64), (1,24)(2,23)(3,22)(4,21)(5,57)(6,60)(7,59)(8,58)(9,19)(10,18)(11,17)(12,20)(13,48)(14,47)(15,46)(16,45)(25,33)(26,36)(27,35)(28,34)(29,39)(30,38)(31,37)(32,40)(41,50)(42,49)(43,52)(44,51)(53,62)(54,61)(55,64)(56,63) );
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)], [(1,45,17,41),(2,46,18,42),(3,47,19,43),(4,48,20,44),(5,38,34,62),(6,39,35,63),(7,40,36,64),(8,37,33,61),(9,52,22,14),(10,49,23,15),(11,50,24,16),(12,51,21,13),(25,54,58,31),(26,55,59,32),(27,56,60,29),(28,53,57,30)], [(1,32,19,53),(2,54,20,29),(3,30,17,55),(4,56,18,31),(5,52,36,16),(6,13,33,49),(7,50,34,14),(8,15,35,51),(9,64,24,38),(10,39,21,61),(11,62,22,40),(12,37,23,63),(25,44,60,46),(26,47,57,41),(27,42,58,48),(28,45,59,43)], [(1,11,17,24),(2,21,18,12),(3,9,19,22),(4,23,20,10),(5,32,34,55),(6,56,35,29),(7,30,36,53),(8,54,33,31),(13,46,51,42),(14,43,52,47),(15,48,49,44),(16,41,50,45),(25,61,58,37),(26,38,59,62),(27,63,60,39),(28,40,57,64)], [(1,24),(2,23),(3,22),(4,21),(5,57),(6,60),(7,59),(8,58),(9,19),(10,18),(11,17),(12,20),(13,48),(14,47),(15,46),(16,45),(25,33),(26,36),(27,35),(28,34),(29,39),(30,38),(31,37),(32,40),(41,50),(42,49),(43,52),(44,51),(53,62),(54,61),(55,64),(56,63)]])
Matrix representation of C42.476C23 ►in GL6(𝔽17)
| 1 | 15 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 7 |
| 0 | 0 | 0 | 16 | 10 | 0 |
| 0 | 0 | 0 | 10 | 1 | 0 |
| 0 | 0 | 7 | 0 | 0 | 1 |
| 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 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 3 | 14 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 1 | 0 |
| 0 | 0 | 10 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 10 |
| 0 | 0 | 0 | 16 | 10 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 |
| 16 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 1 | 0 |
| 0 | 0 | 7 | 0 | 0 | 1 |
| 0 | 0 | 16 | 0 | 0 | 7 |
| 0 | 0 | 0 | 16 | 10 | 0 |
G:=sub<GL(6,GF(17))| [1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,0,0,7,0,0,0,16,10,0,0,0,0,10,1,0,0,0,7,0,0,1],[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],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,14,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,10,1,0,0,0,10,0,0,16,0,0,1,0,0,10,0,0,0,16,10,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,7,16,0,0,0,10,0,0,16,0,0,1,0,0,10,0,0,0,1,7,0] >;
C42.476C23 in GAP, Magma, Sage, TeX
C_4^2._{476}C_2^3 % in TeX
G:=Group("C4^2.476C2^3"); // GroupNames label
G:=SmallGroup(128,2059);
// by ID
G=gap.SmallGroup(128,2059);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,352,2019,346,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,e*a*e=a^-1,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*c,e*d*e=b^2*d>;
// generators/relations
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