p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊35D4, C24.105C23, C23.723C24, C22.3792- 1+4, C22.4962+ 1+4, C42⋊9C4⋊39C2, C23.Q8⋊94C2, (C2×C42).735C22, (C22×C4).234C23, C22.455(C22×D4), C23.10D4.73C2, (C22×D4).298C22, C23.81C23⋊135C2, C2.46(C22.54C24), C2.C42.426C22, C2.61(C22.31C24), C2.62(C22.57C24), (C2×C4).440(C2×D4), (C2×C42⋊2C2)⋊28C2, (C2×C4⋊C4).532C22, (C2×C22⋊C4).341C22, SmallGroup(128,1555)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊35D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
Subgroups: 468 in 228 conjugacy classes, 92 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C22×D4, C42⋊9C4, C23.10D4, C23.Q8, C23.81C23, C2×C42⋊2C2, C42⋊35D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C22.54C24, C22.57C24, C42⋊35D4
Character table of C42⋊35D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | -4 | -4 | 4 | 4 | -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 |
| ρ22 | 4 | 4 | -4 | -4 | -4 | -4 | 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 |
| ρ23 | 4 | -4 | 4 | 4 | -4 | -4 | -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 |
| ρ24 | 4 | 4 | 4 | -4 | -4 | 4 | -4 | -4 | 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 |
| ρ25 | 4 | -4 | -4 | 4 | -4 | 4 | 4 | -4 | 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 | 4 | -4 | -4 | -4 | 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 |
►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 42 53 13)(2 43 54 14)(3 44 55 15)(4 41 56 16)(5 18 40 47)(6 19 37 48)(7 20 38 45)(8 17 39 46)(9 57 22 32)(10 58 23 29)(11 59 24 30)(12 60 21 31)(25 64 51 36)(26 61 52 33)(27 62 49 34)(28 63 50 35)
(1 33 11 19)(2 36 12 18)(3 35 9 17)(4 34 10 20)(5 43 51 60)(6 42 52 59)(7 41 49 58)(8 44 50 57)(13 26 30 37)(14 25 31 40)(15 28 32 39)(16 27 29 38)(21 47 54 64)(22 46 55 63)(23 45 56 62)(24 48 53 61)
(2 56)(4 54)(5 25)(6 50)(7 27)(8 52)(10 21)(12 23)(13 15)(14 43)(16 41)(17 35)(18 62)(19 33)(20 64)(26 39)(28 37)(29 58)(30 32)(31 60)(34 47)(36 45)(38 49)(40 51)(42 44)(46 63)(48 61)(57 59)
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), (1,42,53,13)(2,43,54,14)(3,44,55,15)(4,41,56,16)(5,18,40,47)(6,19,37,48)(7,20,38,45)(8,17,39,46)(9,57,22,32)(10,58,23,29)(11,59,24,30)(12,60,21,31)(25,64,51,36)(26,61,52,33)(27,62,49,34)(28,63,50,35), (1,33,11,19)(2,36,12,18)(3,35,9,17)(4,34,10,20)(5,43,51,60)(6,42,52,59)(7,41,49,58)(8,44,50,57)(13,26,30,37)(14,25,31,40)(15,28,32,39)(16,27,29,38)(21,47,54,64)(22,46,55,63)(23,45,56,62)(24,48,53,61), (2,56)(4,54)(5,25)(6,50)(7,27)(8,52)(10,21)(12,23)(13,15)(14,43)(16,41)(17,35)(18,62)(19,33)(20,64)(26,39)(28,37)(29,58)(30,32)(31,60)(34,47)(36,45)(38,49)(40,51)(42,44)(46,63)(48,61)(57,59)>;
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), (1,42,53,13)(2,43,54,14)(3,44,55,15)(4,41,56,16)(5,18,40,47)(6,19,37,48)(7,20,38,45)(8,17,39,46)(9,57,22,32)(10,58,23,29)(11,59,24,30)(12,60,21,31)(25,64,51,36)(26,61,52,33)(27,62,49,34)(28,63,50,35), (1,33,11,19)(2,36,12,18)(3,35,9,17)(4,34,10,20)(5,43,51,60)(6,42,52,59)(7,41,49,58)(8,44,50,57)(13,26,30,37)(14,25,31,40)(15,28,32,39)(16,27,29,38)(21,47,54,64)(22,46,55,63)(23,45,56,62)(24,48,53,61), (2,56)(4,54)(5,25)(6,50)(7,27)(8,52)(10,21)(12,23)(13,15)(14,43)(16,41)(17,35)(18,62)(19,33)(20,64)(26,39)(28,37)(29,58)(30,32)(31,60)(34,47)(36,45)(38,49)(40,51)(42,44)(46,63)(48,61)(57,59) );
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)], [(1,42,53,13),(2,43,54,14),(3,44,55,15),(4,41,56,16),(5,18,40,47),(6,19,37,48),(7,20,38,45),(8,17,39,46),(9,57,22,32),(10,58,23,29),(11,59,24,30),(12,60,21,31),(25,64,51,36),(26,61,52,33),(27,62,49,34),(28,63,50,35)], [(1,33,11,19),(2,36,12,18),(3,35,9,17),(4,34,10,20),(5,43,51,60),(6,42,52,59),(7,41,49,58),(8,44,50,57),(13,26,30,37),(14,25,31,40),(15,28,32,39),(16,27,29,38),(21,47,54,64),(22,46,55,63),(23,45,56,62),(24,48,53,61)], [(2,56),(4,54),(5,25),(6,50),(7,27),(8,52),(10,21),(12,23),(13,15),(14,43),(16,41),(17,35),(18,62),(19,33),(20,64),(26,39),(28,37),(29,58),(30,32),(31,60),(34,47),(36,45),(38,49),(40,51),(42,44),(46,63),(48,61),(57,59)]])
Matrix representation of C42⋊35D4 ►in GL10(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 | 3 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 |
| 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 | 4 |
G:=sub<GL(10,GF(5))| [4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,3,0,0,2,0,0,0,0,0,0,0,3,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,4,2,2,3,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,3,0,1,3,0,0,0,0,0,0,0,3,3,1],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,0,3,0,0,1,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2],[2,0,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,4,4,0,1,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,1,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,4,2,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4] >;
C42⋊35D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{35}D_4 % in TeX
G:=Group("C4^2:35D4"); // GroupNames label
G:=SmallGroup(128,1555);
// by ID
G=gap.SmallGroup(128,1555);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,794,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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