metabelian, supersoluble, monomial
Aliases: C52⋊2D4, D10⋊1D5, C10.3D10, C2.3D52, (D5×C10)⋊1C2, C5⋊2(C5⋊D4), C52⋊6C4⋊2C2, (C5×C10).3C22, SmallGroup(200,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊2D4
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Character table of C52⋊2D4
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 10O | 10P | |
| 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 | 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 | orthogonal lifted from D4 |
| 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 | -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 |
| ρ8 | 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 |
| ρ9 | 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 |
| 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 | 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 |
| ρ13 | 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 |
| ρ14 | 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 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | 0 | 0 | 0 | 0 | ζ54-ζ5 | complex lifted from C5⋊D4 |
| 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 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | 0 | 0 | 0 | 0 | -ζ54+ζ5 | complex lifted from C5⋊D4 |
| ρ16 | 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 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | 0 | 0 | 0 | 0 | ζ53-ζ52 | complex lifted from C5⋊D4 |
| 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 | -ζ53+ζ52 | ζ54-ζ5 | -ζ54+ζ5 | ζ53-ζ52 | 0 | complex lifted from C5⋊D4 |
| ρ18 | 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 | ζ54-ζ5 | ζ53-ζ52 | -ζ53+ζ52 | -ζ54+ζ5 | 0 | complex lifted from C5⋊D4 |
| ρ19 | 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 | -ζ54+ζ5 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | 0 | complex lifted from C5⋊D4 |
| ρ20 | 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 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | 0 | 0 | 0 | 0 | -ζ53+ζ52 | complex lifted from C5⋊D4 |
| ρ21 | 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 | ζ53-ζ52 | -ζ54+ζ5 | ζ54-ζ5 | -ζ53+ζ52 | 0 | complex lifted from C5⋊D4 |
| 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 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 35 34 33 32)(36 40 39 38 37)
(1 39 6 34)(2 38 7 33)(3 37 8 32)(4 36 9 31)(5 40 10 35)(11 29 16 24)(12 28 17 23)(13 27 18 22)(14 26 19 21)(15 30 20 25)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)
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), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,35,34,33,32)(36,40,39,38,37), (1,39,6,34)(2,38,7,33)(3,37,8,32)(4,36,9,31)(5,40,10,35)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,30,20,25), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)>;
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), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,35,34,33,32)(36,40,39,38,37), (1,39,6,34)(2,38,7,33)(3,37,8,32)(4,36,9,31)(5,40,10,35)(11,29,16,24)(12,28,17,23)(13,27,18,22)(14,26,19,21)(15,30,20,25), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40) );
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)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,35,34,33,32),(36,40,39,38,37)], [(1,39,6,34),(2,38,7,33),(3,37,8,32),(4,36,9,31),(5,40,10,35),(11,29,16,24),(12,28,17,23),(13,27,18,22),(14,26,19,21),(15,30,20,25)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40)]])
C52⋊2D4 is a maximal subgroup of
C52⋊D8 C52⋊SD16 D20⋊5D5 D10.9D10 C20⋊D10 D10.4D10 D5×C5⋊D4
C52⋊2D4 is a maximal quotient of C52⋊2D8 D20.D5 C52⋊2Q16 D10⋊Dic5 C10.Dic10
Matrix representation of C52⋊2D4 ►in GL4(𝔽41) generated by
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 34 |
| 23 | 6 | 0 | 0 |
| 21 | 18 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 23 | 6 | 0 | 0 |
| 35 | 18 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[23,21,0,0,6,18,0,0,0,0,0,1,0,0,1,0],[23,35,0,0,6,18,0,0,0,0,0,1,0,0,1,0] >;
C52⋊2D4 in GAP, Magma, Sage, TeX
C_5^2\rtimes_2D_4
% in TeX
G:=Group("C5^2:2D4"); // GroupNames label
G:=SmallGroup(200,24);
// by ID
G=gap.SmallGroup(200,24);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,61,328,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊2D4 in TeX
Character table of C52⋊2D4 in TeX