p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.42C23, C4.522+ 1+4, C8⋊D4⋊34C2, C8⋊8D4⋊48C2, C8⋊9D4⋊10C2, C4⋊C4.363D4, D4.Q8⋊34C2, D4⋊6D4⋊10C2, D8⋊C4⋊22C2, D4⋊D4⋊42C2, (C2×D4).167D4, C4⋊C8.98C22, C22⋊C4.46D4, D4.21(C4○D4), D4.D4⋊21C2, C4⋊C4.406C23, (C2×C8).352C23, (C2×C4).499C24, (C2×D8).84C22, C23.319(C2×D4), C4⋊Q8.147C22, C8⋊C4.39C22, C2.73(D4○SD16), (C4×D4).152C22, (C2×D4).229C23, C4⋊D4.78C22, C22⋊C8.76C22, (C2×Q8).214C23, C2.135(D4⋊5D4), C2.D8.117C22, C4.Q8.104C22, C22⋊Q8.78C22, C23.24D4⋊30C2, C23.47D4⋊15C2, C23.36D4⋊14C2, C23.20D4⋊31C2, (C22×C8).362C22, Q8⋊C4.68C22, (C2×SD16).99C22, C22.759(C22×D4), C42.C2.37C22, D4⋊C4.186C22, C2.82(D8⋊C22), C22.47C24⋊3C2, (C22×C4).1143C23, C42.30C22⋊8C2, C42⋊C2.186C22, (C2×M4(2)).108C22, C4.224(C2×C4○D4), (C2×C4).596(C2×D4), (C2×C4⋊C4).664C22, (C2×C4○D4).205C22, SmallGroup(128,2039)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.42C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2b2, ab=ba, cac-1=eae=a-1b2, dad=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2b2c, ede=b2d >
Subgroups: 400 in 197 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×C4○D4, C23.24D4, C23.36D4, C8⋊9D4, D8⋊C4, D4⋊D4, D4.D4, C8⋊8D4, C8⋊D4, D4.Q8, C23.47D4, C23.20D4, C42.30C22, D4⋊6D4, C22.47C24, C42.42C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D4⋊5D4, D8⋊C22, D4○SD16, C42.42C23
Character table of C42.42C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 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 | 0 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 0 | -2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 0 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | -2i | 0 | 0 | 2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ24 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | -2 | 0 | 2 | -2 | 0 | 0 | 0 | 2i | 0 | 0 | -2i | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ25 | 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 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 4i | 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 | 0 | 0 | 4i | -4i | 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 | 0 | 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 | 2√-2 | 0 | 0 | -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 60 30 23)(2 57 31 24)(3 58 32 21)(4 59 29 22)(5 40 47 27)(6 37 48 28)(7 38 45 25)(8 39 46 26)(9 63 50 42)(10 64 51 43)(11 61 52 44)(12 62 49 41)(13 18 36 53)(14 19 33 54)(15 20 34 55)(16 17 35 56)
(1 35 32 14)(2 15 29 36)(3 33 30 16)(4 13 31 34)(5 49 45 10)(6 11 46 50)(7 51 47 12)(8 9 48 52)(17 21 54 60)(18 57 55 22)(19 23 56 58)(20 59 53 24)(25 43 40 62)(26 63 37 44)(27 41 38 64)(28 61 39 42)
(1 38)(2 26)(3 40)(4 28)(5 58)(6 22)(7 60)(8 24)(9 15)(10 35)(11 13)(12 33)(14 49)(16 51)(17 64)(18 44)(19 62)(20 42)(21 47)(23 45)(25 30)(27 32)(29 37)(31 39)(34 50)(36 52)(41 54)(43 56)(46 57)(48 59)(53 61)(55 63)
(1 35)(2 15)(3 33)(4 13)(5 62)(6 44)(7 64)(8 42)(9 39)(10 25)(11 37)(12 27)(14 32)(16 30)(17 23)(18 59)(19 21)(20 57)(22 53)(24 55)(26 50)(28 52)(29 36)(31 34)(38 51)(40 49)(41 47)(43 45)(46 63)(48 61)(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), (1,60,30,23)(2,57,31,24)(3,58,32,21)(4,59,29,22)(5,40,47,27)(6,37,48,28)(7,38,45,25)(8,39,46,26)(9,63,50,42)(10,64,51,43)(11,61,52,44)(12,62,49,41)(13,18,36,53)(14,19,33,54)(15,20,34,55)(16,17,35,56), (1,35,32,14)(2,15,29,36)(3,33,30,16)(4,13,31,34)(5,49,45,10)(6,11,46,50)(7,51,47,12)(8,9,48,52)(17,21,54,60)(18,57,55,22)(19,23,56,58)(20,59,53,24)(25,43,40,62)(26,63,37,44)(27,41,38,64)(28,61,39,42), (1,38)(2,26)(3,40)(4,28)(5,58)(6,22)(7,60)(8,24)(9,15)(10,35)(11,13)(12,33)(14,49)(16,51)(17,64)(18,44)(19,62)(20,42)(21,47)(23,45)(25,30)(27,32)(29,37)(31,39)(34,50)(36,52)(41,54)(43,56)(46,57)(48,59)(53,61)(55,63), (1,35)(2,15)(3,33)(4,13)(5,62)(6,44)(7,64)(8,42)(9,39)(10,25)(11,37)(12,27)(14,32)(16,30)(17,23)(18,59)(19,21)(20,57)(22,53)(24,55)(26,50)(28,52)(29,36)(31,34)(38,51)(40,49)(41,47)(43,45)(46,63)(48,61)(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), (1,60,30,23)(2,57,31,24)(3,58,32,21)(4,59,29,22)(5,40,47,27)(6,37,48,28)(7,38,45,25)(8,39,46,26)(9,63,50,42)(10,64,51,43)(11,61,52,44)(12,62,49,41)(13,18,36,53)(14,19,33,54)(15,20,34,55)(16,17,35,56), (1,35,32,14)(2,15,29,36)(3,33,30,16)(4,13,31,34)(5,49,45,10)(6,11,46,50)(7,51,47,12)(8,9,48,52)(17,21,54,60)(18,57,55,22)(19,23,56,58)(20,59,53,24)(25,43,40,62)(26,63,37,44)(27,41,38,64)(28,61,39,42), (1,38)(2,26)(3,40)(4,28)(5,58)(6,22)(7,60)(8,24)(9,15)(10,35)(11,13)(12,33)(14,49)(16,51)(17,64)(18,44)(19,62)(20,42)(21,47)(23,45)(25,30)(27,32)(29,37)(31,39)(34,50)(36,52)(41,54)(43,56)(46,57)(48,59)(53,61)(55,63), (1,35)(2,15)(3,33)(4,13)(5,62)(6,44)(7,64)(8,42)(9,39)(10,25)(11,37)(12,27)(14,32)(16,30)(17,23)(18,59)(19,21)(20,57)(22,53)(24,55)(26,50)(28,52)(29,36)(31,34)(38,51)(40,49)(41,47)(43,45)(46,63)(48,61)(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)], [(1,60,30,23),(2,57,31,24),(3,58,32,21),(4,59,29,22),(5,40,47,27),(6,37,48,28),(7,38,45,25),(8,39,46,26),(9,63,50,42),(10,64,51,43),(11,61,52,44),(12,62,49,41),(13,18,36,53),(14,19,33,54),(15,20,34,55),(16,17,35,56)], [(1,35,32,14),(2,15,29,36),(3,33,30,16),(4,13,31,34),(5,49,45,10),(6,11,46,50),(7,51,47,12),(8,9,48,52),(17,21,54,60),(18,57,55,22),(19,23,56,58),(20,59,53,24),(25,43,40,62),(26,63,37,44),(27,41,38,64),(28,61,39,42)], [(1,38),(2,26),(3,40),(4,28),(5,58),(6,22),(7,60),(8,24),(9,15),(10,35),(11,13),(12,33),(14,49),(16,51),(17,64),(18,44),(19,62),(20,42),(21,47),(23,45),(25,30),(27,32),(29,37),(31,39),(34,50),(36,52),(41,54),(43,56),(46,57),(48,59),(53,61),(55,63)], [(1,35),(2,15),(3,33),(4,13),(5,62),(6,44),(7,64),(8,42),(9,39),(10,25),(11,37),(12,27),(14,32),(16,30),(17,23),(18,59),(19,21),(20,57),(22,53),(24,55),(26,50),(28,52),(29,36),(31,34),(38,51),(40,49),(41,47),(43,45),(46,63),(48,61),(54,58),(56,60)]])
Matrix representation of C42.42C23 ►in GL6(𝔽17)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 1 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 7 |
| 0 | 0 | 5 | 14 | 12 | 12 |
| 0 | 0 | 14 | 5 | 3 | 12 |
| 0 | 0 | 2 | 0 | 0 | 15 |
| 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 | 0 | 16 | 0 | 1 |
| 0 | 0 | 1 | 16 | 16 | 0 |
| 16 | 8 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 7 | 11 | 0 |
| 0 | 0 | 3 | 12 | 14 | 14 |
| 0 | 0 | 3 | 12 | 14 | 5 |
| 0 | 0 | 0 | 15 | 2 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 1 | 4 | 4 |
| 0 | 0 | 7 | 0 | 0 | 4 |
| 0 | 0 | 13 | 11 | 11 | 10 |
| 0 | 0 | 7 | 7 | 10 | 10 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 10 | 6 | 0 |
| 0 | 0 | 12 | 5 | 3 | 14 |
| 0 | 0 | 12 | 5 | 3 | 5 |
| 0 | 0 | 0 | 2 | 15 | 0 |
G:=sub<GL(6,GF(17))| [4,1,0,0,0,0,0,13,0,0,0,0,0,0,2,5,14,2,0,0,0,14,5,0,0,0,0,12,3,0,0,0,7,12,12,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[16,4,0,0,0,0,8,1,0,0,0,0,0,0,8,3,3,0,0,0,7,12,12,15,0,0,11,14,14,2,0,0,0,14,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,7,13,7,0,0,1,0,11,7,0,0,4,0,11,10,0,0,4,4,10,10],[1,0,0,0,0,0,9,16,0,0,0,0,0,0,9,12,12,0,0,0,10,5,5,2,0,0,6,3,3,15,0,0,0,14,5,0] >;
C42.42C23 in GAP, Magma, Sage, TeX
C_4^2._{42}C_2^3 % in TeX
G:=Group("C4^2.42C2^3"); // GroupNames label
G:=SmallGroup(128,2039);
// by ID
G=gap.SmallGroup(128,2039);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,346,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2*b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1*b^2,d*a*d=a*b^2,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
// generators/relations
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