p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊34D4, C23.719C24, C24.102C23, C22.4922+ 1+4, C22.3752- 1+4, C42⋊8C4⋊72C2, C23⋊2D4⋊49C2, C23⋊Q8⋊62C2, (C2×C42).731C22, (C22×C4).230C23, C22.451(C22×D4), C23.10D4⋊110C2, (C22×D4).296C22, (C22×Q8).234C22, C2.71(C22.29C24), C2.16(C24⋊C22), C2.C42.422C22, C2.51(C22.56C24), C2.58(C22.31C24), (C2×C4).436(C2×D4), (C2×C4.4D4)⋊33C2, (C2×C4⋊C4).528C22, (C2×C22⋊C4).338C22, SmallGroup(128,1551)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊34D4
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
Subgroups: 708 in 290 conjugacy classes, 92 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C22×D4, C22×Q8, C42⋊8C4, C23⋊2D4, C23⋊Q8, C23.10D4, C2×C4.4D4, C42⋊34D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C22.29C24, C22.31C24, C24⋊C22, C22.56C24, C42⋊34D4
Character table of C42⋊34D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | 0 | 2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | 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 | orthogonal lifted from 2+ 1+4 |
| 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 | orthogonal lifted from 2+ 1+4 |
| 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 36 59 9)(2 33 60 10)(3 34 57 11)(4 35 58 12)(5 28 44 32)(6 25 41 29)(7 26 42 30)(8 27 43 31)(13 23 51 53)(14 24 52 54)(15 21 49 55)(16 22 50 56)(17 45 39 61)(18 46 40 62)(19 47 37 63)(20 48 38 64)
(1 46 15 26)(2 45 16 25)(3 48 13 28)(4 47 14 27)(5 36 20 21)(6 35 17 24)(7 34 18 23)(8 33 19 22)(9 38 55 44)(10 37 56 43)(11 40 53 42)(12 39 54 41)(29 60 61 50)(30 59 62 49)(31 58 63 52)(32 57 64 51)
(1 35)(2 9)(3 33)(4 11)(5 61)(6 46)(7 63)(8 48)(10 57)(12 59)(13 22)(14 53)(15 24)(16 55)(17 26)(18 31)(19 28)(20 29)(21 50)(23 52)(25 38)(27 40)(30 39)(32 37)(34 58)(36 60)(41 62)(42 47)(43 64)(44 45)(49 54)(51 56)
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), (1,36,59,9)(2,33,60,10)(3,34,57,11)(4,35,58,12)(5,28,44,32)(6,25,41,29)(7,26,42,30)(8,27,43,31)(13,23,51,53)(14,24,52,54)(15,21,49,55)(16,22,50,56)(17,45,39,61)(18,46,40,62)(19,47,37,63)(20,48,38,64), (1,46,15,26)(2,45,16,25)(3,48,13,28)(4,47,14,27)(5,36,20,21)(6,35,17,24)(7,34,18,23)(8,33,19,22)(9,38,55,44)(10,37,56,43)(11,40,53,42)(12,39,54,41)(29,60,61,50)(30,59,62,49)(31,58,63,52)(32,57,64,51), (1,35)(2,9)(3,33)(4,11)(5,61)(6,46)(7,63)(8,48)(10,57)(12,59)(13,22)(14,53)(15,24)(16,55)(17,26)(18,31)(19,28)(20,29)(21,50)(23,52)(25,38)(27,40)(30,39)(32,37)(34,58)(36,60)(41,62)(42,47)(43,64)(44,45)(49,54)(51,56)>;
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), (1,36,59,9)(2,33,60,10)(3,34,57,11)(4,35,58,12)(5,28,44,32)(6,25,41,29)(7,26,42,30)(8,27,43,31)(13,23,51,53)(14,24,52,54)(15,21,49,55)(16,22,50,56)(17,45,39,61)(18,46,40,62)(19,47,37,63)(20,48,38,64), (1,46,15,26)(2,45,16,25)(3,48,13,28)(4,47,14,27)(5,36,20,21)(6,35,17,24)(7,34,18,23)(8,33,19,22)(9,38,55,44)(10,37,56,43)(11,40,53,42)(12,39,54,41)(29,60,61,50)(30,59,62,49)(31,58,63,52)(32,57,64,51), (1,35)(2,9)(3,33)(4,11)(5,61)(6,46)(7,63)(8,48)(10,57)(12,59)(13,22)(14,53)(15,24)(16,55)(17,26)(18,31)(19,28)(20,29)(21,50)(23,52)(25,38)(27,40)(30,39)(32,37)(34,58)(36,60)(41,62)(42,47)(43,64)(44,45)(49,54)(51,56) );
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)], [(1,36,59,9),(2,33,60,10),(3,34,57,11),(4,35,58,12),(5,28,44,32),(6,25,41,29),(7,26,42,30),(8,27,43,31),(13,23,51,53),(14,24,52,54),(15,21,49,55),(16,22,50,56),(17,45,39,61),(18,46,40,62),(19,47,37,63),(20,48,38,64)], [(1,46,15,26),(2,45,16,25),(3,48,13,28),(4,47,14,27),(5,36,20,21),(6,35,17,24),(7,34,18,23),(8,33,19,22),(9,38,55,44),(10,37,56,43),(11,40,53,42),(12,39,54,41),(29,60,61,50),(30,59,62,49),(31,58,63,52),(32,57,64,51)], [(1,35),(2,9),(3,33),(4,11),(5,61),(6,46),(7,63),(8,48),(10,57),(12,59),(13,22),(14,53),(15,24),(16,55),(17,26),(18,31),(19,28),(20,29),(21,50),(23,52),(25,38),(27,40),(30,39),(32,37),(34,58),(36,60),(41,62),(42,47),(43,64),(44,45),(49,54),(51,56)]])
Matrix representation of C42⋊34D4 ►in GL10(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 4 | 2 |
| 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 | 0 | 4 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 4 | 0 | 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 | 4 | 1 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | 0 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 4 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 4 | 2 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 | 1 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 1 | 0 |
G:=sub<GL(10,GF(5))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,0,0,0,0,1,2,4,3,0,0,0,0,0,0,3,2,4,0,0,0,0,0,0,0,4,1,0,4,0,0,0,0,0,0,2,0,1,2],[4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,1],[0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,1,3,2,4,0,0,0,0,0,0,3,3,2,0,0,0,0,0,0,0,4,4,4,4,0,0,0,0,0,0,2,0,1,2],[4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,0,2,0,2,0,0,0,0,0,0,4,0,2,0,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,0,0,0,0,2,4,4,0,0,0,0,0,0,0,2,2,2,4,0,0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,0,3,4,0] >;
C42⋊34D4 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{34}D_4 % in TeX
G:=Group("C4^2:34D4"); // GroupNames label
G:=SmallGroup(128,1551);
// by ID
G=gap.SmallGroup(128,1551);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,758,723,794,185,80]);
// 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*b^2,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations
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