non-abelian, soluble, monomial
Aliases: C52⋊D8, C2.3D5≀C2, (C5×C10).3D4, C52⋊5C8⋊3C2, C52⋊2D4⋊5C2, C52⋊6C4.6C22, SmallGroup(400,131)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊D8
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=dbd=a2, dad=cbc-1=b3, dcd=c-1 >
20C2
20C2
2C5
2C5
2C5
10C22
10C22
25C4
2C10
2C10
2C10
4D5
4D5
20C10
20C10
25D4
25D4
25C8
2D10
2D10
10Dic5
10C2×C10
10Dic5
10C2×C10
10Dic5
25D8
10C5⋊D4
10C5⋊D4
10C5⋊C8
Character table of C52⋊D8
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 5C | 5D | 5E | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | |
| size | 1 | 1 | 20 | 20 | 50 | 4 | 4 | 4 | 4 | 8 | 50 | 50 | 4 | 4 | 4 | 4 | 8 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ6 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | √2 | -√2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ7 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -√2 | √2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ8 | 4 | 4 | 0 | -2 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | 0 | 1-√5/2 | 0 | 1+√5/2 | 0 | 0 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ9 | 4 | 4 | -2 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | 1+√5/2 | 0 | 1-√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ10 | 4 | 4 | 2 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | -1-√5/2 | 0 | -1+√5/2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ11 | 4 | 4 | 0 | 2 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | 0 | -1+√5/2 | 0 | -1-√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ12 | 4 | 4 | -2 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | 1-√5/2 | 0 | 1+√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ13 | 4 | 4 | 0 | 2 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | 0 | -1-√5/2 | 0 | -1+√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ14 | 4 | 4 | 0 | -2 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | 0 | 1+√5/2 | 0 | 1-√5/2 | 0 | 0 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ15 | 4 | 4 | 2 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | -1+√5/2 | 0 | -1-√5/2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | 0 | 0 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 1 | ζ53-ζ52 | 0 | ζ54-ζ5 | 0 | -ζ53+ζ52 | -ζ54+ζ5 | 0 | 0 | complex faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | 0 | 0 | -3+√5/2 | -3-√5/2 | 1-√5 | 1+√5 | 1 | 0 | ζ53-ζ52 | 0 | ζ54-ζ5 | 0 | 0 | -ζ54+ζ5 | -ζ53+ζ52 | complex faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | 0 | 0 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 1 | -ζ54+ζ5 | 0 | ζ53-ζ52 | 0 | ζ54-ζ5 | -ζ53+ζ52 | 0 | 0 | complex faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3-√5/2 | 3+√5/2 | -1 | 0 | 0 | 1+√5 | 1-√5 | -3+√5/2 | -3-√5/2 | 1 | -ζ53+ζ52 | 0 | -ζ54+ζ5 | 0 | ζ53-ζ52 | ζ54-ζ5 | 0 | 0 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | 0 | 0 | -3-√5/2 | -3+√5/2 | 1+√5 | 1-√5 | 1 | 0 | ζ54-ζ5 | 0 | -ζ53+ζ52 | 0 | 0 | ζ53-ζ52 | -ζ54+ζ5 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1-√5 | -1+√5 | -1 | 0 | 0 | -3-√5/2 | -3+√5/2 | 1+√5 | 1-√5 | 1 | 0 | -ζ54+ζ5 | 0 | ζ53-ζ52 | 0 | 0 | -ζ53+ζ52 | ζ54-ζ5 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3+√5/2 | 3-√5/2 | -1 | 0 | 0 | 1-√5 | 1+√5 | -3-√5/2 | -3+√5/2 | 1 | ζ54-ζ5 | 0 | -ζ53+ζ52 | 0 | -ζ54+ζ5 | ζ53-ζ52 | 0 | 0 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1+√5 | -1-√5 | -1 | 0 | 0 | -3+√5/2 | -3-√5/2 | 1-√5 | 1+√5 | 1 | 0 | -ζ53+ζ52 | 0 | -ζ54+ζ5 | 0 | 0 | ζ54-ζ5 | ζ53-ζ52 | complex faithful |
| ρ24 | 8 | 8 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ25 | 8 | -8 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | 0 | 0 | 2 | 2 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 40 points
Generators in S40
(1 33 18 9 30)(2 19 31 34 10)(3 32 11 20 35)(4 12 36 25 21)(5 37 22 13 26)(6 23 27 38 14)(7 28 15 24 39)(8 16 40 29 17)
(1 18 30 33 9)(2 34 19 10 31)(3 11 35 32 20)(4 25 12 21 36)(5 22 26 37 13)(6 38 23 14 27)(7 15 39 28 24)(8 29 16 17 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)
(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 19)(20 24)(21 23)(25 27)(28 32)(29 31)(34 40)(35 39)(36 38)
G:=sub<Sym(40)| (1,33,18,9,30)(2,19,31,34,10)(3,32,11,20,35)(4,12,36,25,21)(5,37,22,13,26)(6,23,27,38,14)(7,28,15,24,39)(8,16,40,29,17), (1,18,30,33,9)(2,34,19,10,31)(3,11,35,32,20)(4,25,12,21,36)(5,22,26,37,13)(6,38,23,14,27)(7,15,39,28,24)(8,29,16,17,40), (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38)>;
G:=Group( (1,33,18,9,30)(2,19,31,34,10)(3,32,11,20,35)(4,12,36,25,21)(5,37,22,13,26)(6,23,27,38,14)(7,28,15,24,39)(8,16,40,29,17), (1,18,30,33,9)(2,34,19,10,31)(3,11,35,32,20)(4,25,12,21,36)(5,22,26,37,13)(6,38,23,14,27)(7,15,39,28,24)(8,29,16,17,40), (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), (2,8)(3,7)(4,6)(10,16)(11,15)(12,14)(17,19)(20,24)(21,23)(25,27)(28,32)(29,31)(34,40)(35,39)(36,38) );
G=PermutationGroup([[(1,33,18,9,30),(2,19,31,34,10),(3,32,11,20,35),(4,12,36,25,21),(5,37,22,13,26),(6,23,27,38,14),(7,28,15,24,39),(8,16,40,29,17)], [(1,18,30,33,9),(2,34,19,10,31),(3,11,35,32,20),(4,25,12,21,36),(5,22,26,37,13),(6,38,23,14,27),(7,15,39,28,24),(8,29,16,17,40)], [(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)], [(2,8),(3,7),(4,6),(10,16),(11,15),(12,14),(17,19),(20,24),(21,23),(25,27),(28,32),(29,31),(34,40),(35,39),(36,38)]])
Matrix representation of C52⋊D8 ►in GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 37 |
| 10 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,18,0,0,0,0,10,0,0,0,0,37],[10,0,0,0,0,37,0,0,0,0,16,0,0,0,0,18],[0,0,1,0,0,0,0,40,0,1,0,0,1,0,0,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
C52⋊D8 in GAP, Magma, Sage, TeX
C_5^2\rtimes D_8
% in TeX
G:=Group("C5^2:D8"); // GroupNames label
G:=SmallGroup(400,131);
// by ID
G=gap.SmallGroup(400,131);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,5,73,218,116,50,7204,1210,496,1157,299,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^2,d*a*d=c*b*c^-1=b^3,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊D8 in TeX
Character table of C52⋊D8 in TeX