non-abelian, soluble, monomial
Aliases: C52⋊SD16, C2.4D5≀C2, (C5×C10).4D4, C52⋊2D4.C2, C52⋊5C8⋊4C2, C52⋊2Q8⋊1C2, C52⋊6C4.7C22, SmallGroup(400,132)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊SD16
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=a3, dad=b3, cbc-1=b2, dbd=a2, dcd=c3 >
20C2
2C5
2C5
2C5
10C22
10C4
25C4
2C10
2C10
2C10
4D5
20C10
25C8
25Q8
25D4
2D10
2Dic5
10Dic5
10C20
10Dic5
10Dic5
10C2×C10
25SD16
10Dic10
10C5⋊D4
10C5⋊C8
Character table of C52⋊SD16
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 20A | 20B | 20C | 20D | |
| 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 | complex lifted from SD16 |
| 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 | complex lifted from SD16 |
| ρ8 | 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 | -1-√5/2 | -1+√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | 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 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ10 | 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 | 0 | 0 | 0 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 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 | 0 | 0 | 0 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ12 | 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 | 0 | 0 | 0 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 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 | 0 | 0 | 0 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ14 | 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 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 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 | -1+√5/2 | -1-√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ16 | 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 | 0 | 0 | 0 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | symplectic faithful, Schur index 2 |
| 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 | 0 | 0 | 0 | ζ43ζ54-ζ43ζ5 | -ζ43ζ54+ζ43ζ5 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | symplectic faithful, Schur index 2 |
| ρ18 | 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 | 0 | 0 | 0 | -ζ4ζ53+ζ4ζ52 | ζ4ζ53-ζ4ζ52 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | symplectic faithful, Schur index 2 |
| ρ19 | 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 | 0 | 0 | 0 | -ζ43ζ54+ζ43ζ5 | ζ43ζ54-ζ43ζ5 | ζ4ζ53-ζ4ζ52 | -ζ4ζ53+ζ4ζ52 | symplectic faithful, Schur index 2 |
| ρ20 | 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 | ζ54-ζ5 | -ζ53+ζ52 | ζ53-ζ52 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 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 | -ζ53+ζ52 | -ζ54+ζ5 | ζ54-ζ5 | 0 | 0 | 0 | 0 | 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 | -ζ53+ζ52 | ζ53-ζ52 | ζ54-ζ5 | -ζ54+ζ5 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 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 | -ζ54+ζ5 | ζ53-ζ52 | -ζ53+ζ52 | 0 | 0 | 0 | 0 | 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 18 35 29 12)(2 30 19 13 36)(3 14 31 37 20)(4 38 15 21 32)(5 22 39 25 16)(6 26 23 9 40)(7 10 27 33 24)(8 34 11 17 28)
(1 35 12 18 29)(2 13 30 36 19)(3 31 20 14 37)(4 21 38 32 15)(5 39 16 22 25)(6 9 26 40 23)(7 27 24 10 33)(8 17 34 28 11)
(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 4)(3 7)(6 8)(9 11)(10 14)(13 15)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)
G:=sub<Sym(40)| (1,18,35,29,12)(2,30,19,13,36)(3,14,31,37,20)(4,38,15,21,32)(5,22,39,25,16)(6,26,23,9,40)(7,10,27,33,24)(8,34,11,17,28), (1,35,12,18,29)(2,13,30,36,19)(3,31,20,14,37)(4,21,38,32,15)(5,39,16,22,25)(6,9,26,40,23)(7,27,24,10,33)(8,17,34,28,11), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)>;
G:=Group( (1,18,35,29,12)(2,30,19,13,36)(3,14,31,37,20)(4,38,15,21,32)(5,22,39,25,16)(6,26,23,9,40)(7,10,27,33,24)(8,34,11,17,28), (1,35,12,18,29)(2,13,30,36,19)(3,31,20,14,37)(4,21,38,32,15)(5,39,16,22,25)(6,9,26,40,23)(7,27,24,10,33)(8,17,34,28,11), (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), (2,4)(3,7)(6,8)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38) );
G=PermutationGroup([[(1,18,35,29,12),(2,30,19,13,36),(3,14,31,37,20),(4,38,15,21,32),(5,22,39,25,16),(6,26,23,9,40),(7,10,27,33,24),(8,34,11,17,28)], [(1,35,12,18,29),(2,13,30,36,19),(3,31,20,14,37),(4,21,38,32,15),(5,39,16,22,25),(6,9,26,40,23),(7,27,24,10,33),(8,17,34,28,11)], [(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)], [(2,4),(3,7),(6,8),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38)]])
Matrix representation of C52⋊SD16 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 10 |
| 37 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,16,0,0,0,0,37,0,0,0,0,10],[37,0,0,0,0,10,0,0,0,0,18,0,0,0,0,16],[0,0,0,40,0,0,40,0,1,0,0,0,0,40,0,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
C52⋊SD16 in GAP, Magma, Sage, TeX
C_5^2\rtimes {\rm SD}_{16} % in TeX
G:=Group("C5^2:SD16"); // GroupNames label
G:=SmallGroup(400,132);
// by ID
G=gap.SmallGroup(400,132);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,5,73,55,218,116,50,7204,1210,496,1157,299,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=a^3,d*a*d=b^3,c*b*c^-1=b^2,d*b*d=a^2,d*c*d=c^3>;
// generators/relations
Export
Subgroup lattice of C52⋊SD16 in TeX
Character table of C52⋊SD16 in TeX