metabelian, supersoluble, monomial
Aliases: C52⋊2Q8, C5⋊1Dic10, C10.5D10, Dic5.1D5, C2.5D52, (C5×C10).5C22, C52⋊6C4.1C2, (C5×Dic5).2C2, SmallGroup(200,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊2Q8
G = < a,b,c,d | a5=b5=c4=1, d2=c2, ab=ba, cac-1=a-1, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Character table of C52⋊2Q8
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 1 | 10 | 10 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | |
| 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 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | -1+√5/2 | orthogonal lifted from D5 |
| ρ6 | 2 | 2 | 0 | 2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 0 | orthogonal lifted from D5 |
| ρ7 | 2 | 2 | 0 | -2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 0 | orthogonal lifted from D10 |
| ρ8 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 1+√5/2 | orthogonal lifted from D10 |
| ρ9 | 2 | 2 | 0 | -2 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 0 | orthogonal lifted from D10 |
| ρ10 | 2 | 2 | 0 | 2 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 0 | orthogonal lifted from D5 |
| ρ11 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | -1-√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | -2 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 1-√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -2 | -2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | -ζ43ζ54+ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | 0 | symplectic lifted from Dic10, Schur index 2 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -2 | -2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | 0 | 0 | 0 | 0 | ζ43ζ54-ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| ρ16 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -2 | -2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | -ζ4ζ53+ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | 0 | symplectic lifted from Dic10, Schur index 2 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -2 | -2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | ζ43ζ54-ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | 0 | symplectic lifted from Dic10, Schur index 2 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -2 | -2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | 0 | 0 | 0 | 0 | ζ4ζ53-ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -2 | -2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | 0 | 0 | 0 | 0 | -ζ4ζ53+ζ4ζ52 | symplectic lifted from Dic10, Schur index 2 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -2 | -2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | ζ4ζ53-ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | 0 | symplectic lifted from Dic10, Schur index 2 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -2 | -2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | 0 | 0 | 0 | 0 | -ζ43ζ54+ζ43ζ5 | symplectic lifted from Dic10, Schur index 2 |
| ρ22 | 4 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1+√5 | -1-√5 | -1 | 3+√5/2 | -1 | 3-√5/2 | -1+√5 | -1-√5 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D52 |
| ρ23 | 4 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1-√5 | -1+√5 | -1 | 3-√5/2 | -1 | 3+√5/2 | -1-√5 | -1+√5 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D52 |
| ρ24 | 4 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1-√5 | -1+√5 | 3-√5/2 | -1 | 3+√5/2 | -1 | -1+√5 | -1-√5 | -1-√5 | -1+√5 | -1 | 3-√5/2 | 3+√5/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D52 |
| ρ25 | 4 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1+√5 | -1-√5 | 3+√5/2 | -1 | 3-√5/2 | -1 | -1-√5 | -1+√5 | -1+√5 | -1-√5 | -1 | 3+√5/2 | 3-√5/2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D52 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1-√5 | -1+√5 | -1 | 3-√5/2 | -1 | 3+√5/2 | 1+√5 | 1-√5 | 1+√5 | 1-√5 | -3-√5/2 | 1 | 1 | -3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1-√5 | -1+√5 | 3-√5/2 | -1 | 3+√5/2 | -1 | 1-√5 | 1+√5 | 1+√5 | 1-√5 | 1 | -3+√5/2 | -3-√5/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | -1+√5 | -1-√5 | -1 | 3+√5/2 | -1 | 3-√5/2 | 1-√5 | 1+√5 | 1-√5 | 1+√5 | -3+√5/2 | 1 | 1 | -3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | -1+√5 | -1-√5 | 3+√5/2 | -1 | 3-√5/2 | -1 | 1+√5 | 1-√5 | 1-√5 | 1+√5 | 1 | -3-√5/2 | -3+√5/2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 40 points
Generators in S40
(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)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)(21 24 22 25 23)(26 29 27 30 28)(31 33 35 32 34)(36 38 40 37 39)
(1 17 8 12)(2 16 9 11)(3 20 10 15)(4 19 6 14)(5 18 7 13)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 35 30 40)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)
G:=sub<Sym(40)| (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), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39), (1,17,8,12)(2,16,9,11)(3,20,10,15)(4,19,6,14)(5,18,7,13)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,35,30,40), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)>;
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), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18)(21,24,22,25,23)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39), (1,17,8,12)(2,16,9,11)(3,20,10,15)(4,19,6,14)(5,18,7,13)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,35,30,40), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35) );
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)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18),(21,24,22,25,23),(26,29,27,30,28),(31,33,35,32,34),(36,38,40,37,39)], [(1,17,8,12),(2,16,9,11),(3,20,10,15),(4,19,6,14),(5,18,7,13),(21,34,26,39),(22,33,27,38),(23,32,28,37),(24,31,29,36),(25,35,30,40)], [(1,28,8,23),(2,29,9,24),(3,30,10,25),(4,26,6,21),(5,27,7,22),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35)]])
C52⋊2Q8 is a maximal subgroup of
C52⋊SD16 C52⋊Q16 D5×Dic10 Dic10⋊D5 D10.9D10 Dic5.D10 D10.4D10
C52⋊2Q8 is a maximal quotient of Dic5⋊Dic5 C10.Dic10
Matrix representation of C52⋊2Q8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 1 | 0 |
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 2 | 13 | 0 | 0 |
| 28 | 39 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 34 | 40 |
| 18 | 21 | 0 | 0 |
| 6 | 23 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,34,1,0,0,40,0],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[2,28,0,0,13,39,0,0,0,0,1,34,0,0,0,40],[18,6,0,0,21,23,0,0,0,0,1,0,0,0,0,1] >;
C52⋊2Q8 in GAP, Magma, Sage, TeX
C_5^2\rtimes_2Q_8
% in TeX
G:=Group("C5^2:2Q8"); // GroupNames label
G:=SmallGroup(200,26);
// by ID
G=gap.SmallGroup(200,26);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,61,26,328,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊2Q8 in TeX
Character table of C52⋊2Q8 in TeX