non-abelian, supersoluble, monomial
Aliases: He5⋊4C4, C52⋊4F5, He5⋊C2.1C2, C5.2(C52⋊C4), Aut(5- 1+2), SmallGroup(500,25)
Series: Derived ►Chief ►Lower central ►Upper central
| He5 — He5⋊4C4 |
Generators and relations for He5⋊4C4
G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=a2b-1c, bc=cb, bd=db, dcd-1=c3 >
Character table of He5⋊4C4
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 5I | 5J | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 25 | 25 | 25 | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 20 | 20 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | 25 | |
| 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 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | -i | -i | -i | i | i | i | i | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | i | i | i | -i | -i | -i | -i | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F5 |
| ρ6 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F5 |
| ρ7 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C4 |
| ρ8 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C4 |
| ρ9 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C4 |
| ρ10 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C52⋊C4 |
| ρ11 | 5 | 1 | -1 | -1 | 5ζ52 | 5ζ53 | 5ζ5 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ5 | -ζ54 | -ζ52 | -ζ53 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | complex faithful |
| ρ12 | 5 | 1 | 1 | 1 | 5ζ52 | 5ζ53 | 5ζ5 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | ζ5 | ζ52 | ζ53 | ζ54 | ζ5 | ζ54 | ζ52 | ζ53 | ζ53 | ζ5 | ζ54 | ζ52 | complex faithful |
| ρ13 | 5 | 1 | 1 | 1 | 5ζ54 | 5ζ5 | 5ζ52 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | ζ52 | ζ54 | ζ5 | ζ53 | ζ52 | ζ53 | ζ54 | ζ5 | ζ5 | ζ52 | ζ53 | ζ54 | complex faithful |
| ρ14 | 5 | 1 | 1 | 1 | 5ζ53 | 5ζ52 | 5ζ54 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54 | ζ53 | ζ52 | ζ5 | ζ54 | ζ5 | ζ53 | ζ52 | ζ52 | ζ54 | ζ5 | ζ53 | complex faithful |
| ρ15 | 5 | 1 | -1 | -1 | 5ζ5 | 5ζ54 | 5ζ53 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ53 | -ζ52 | -ζ5 | -ζ54 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | complex faithful |
| ρ16 | 5 | 1 | 1 | 1 | 5ζ5 | 5ζ54 | 5ζ53 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53 | ζ5 | ζ54 | ζ52 | ζ53 | ζ52 | ζ5 | ζ54 | ζ54 | ζ53 | ζ52 | ζ5 | complex faithful |
| ρ17 | 5 | 1 | -1 | -1 | 5ζ53 | 5ζ52 | 5ζ54 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ54 | -ζ5 | -ζ53 | -ζ52 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | complex faithful |
| ρ18 | 5 | 1 | -1 | -1 | 5ζ54 | 5ζ5 | 5ζ52 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ52 | -ζ53 | -ζ54 | -ζ5 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | complex faithful |
| ρ19 | 5 | -1 | -i | i | 5ζ54 | 5ζ5 | 5ζ52 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ4ζ52 | ζ4ζ53 | ζ4ζ54 | ζ4ζ5 | ζ43ζ5 | ζ43ζ52 | ζ43ζ53 | ζ43ζ54 | complex faithful |
| ρ20 | 5 | -1 | -i | i | 5ζ53 | 5ζ52 | 5ζ54 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ4ζ54 | ζ4ζ5 | ζ4ζ53 | ζ4ζ52 | ζ43ζ52 | ζ43ζ54 | ζ43ζ5 | ζ43ζ53 | complex faithful |
| ρ21 | 5 | -1 | i | -i | 5ζ53 | 5ζ52 | 5ζ54 | 5ζ5 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ43ζ54 | ζ43ζ5 | ζ43ζ53 | ζ43ζ52 | ζ4ζ52 | ζ4ζ54 | ζ4ζ5 | ζ4ζ53 | complex faithful |
| ρ22 | 5 | -1 | i | -i | 5ζ54 | 5ζ5 | 5ζ52 | 5ζ53 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ43ζ52 | ζ43ζ53 | ζ43ζ54 | ζ43ζ5 | ζ4ζ5 | ζ4ζ52 | ζ4ζ53 | ζ4ζ54 | complex faithful |
| ρ23 | 5 | -1 | -i | i | 5ζ52 | 5ζ53 | 5ζ5 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ4ζ5 | ζ4ζ54 | ζ4ζ52 | ζ4ζ53 | ζ43ζ53 | ζ43ζ5 | ζ43ζ54 | ζ43ζ52 | complex faithful |
| ρ24 | 5 | -1 | i | -i | 5ζ52 | 5ζ53 | 5ζ5 | 5ζ54 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ43ζ5 | ζ43ζ54 | ζ43ζ52 | ζ43ζ53 | ζ4ζ53 | ζ4ζ5 | ζ4ζ54 | ζ4ζ52 | complex faithful |
| ρ25 | 5 | -1 | i | -i | 5ζ5 | 5ζ54 | 5ζ53 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ43ζ53 | ζ43ζ52 | ζ43ζ5 | ζ43ζ54 | ζ4ζ54 | ζ4ζ53 | ζ4ζ52 | ζ4ζ5 | complex faithful |
| ρ26 | 5 | -1 | -i | i | 5ζ5 | 5ζ54 | 5ζ53 | 5ζ52 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ4ζ53 | ζ4ζ52 | ζ4ζ5 | ζ4ζ54 | ζ43ζ54 | ζ43ζ53 | ζ43ζ52 | ζ43ζ5 | complex faithful |
►On 25 points - transitive group 25T35
Generators in S25
(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)
(1 13 21 18 9)(2 14 22 19 10)(3 15 23 20 6)(4 11 24 16 7)(5 12 25 17 8)
(2 10 19 22 14)(3 20 15 6 23)(4 24 7 11 16)(5 12 25 17 8)
(2 4 8 23)(3 19 16 25)(5 20 14 11)(6 22 24 12)(7 17 15 10)
G:=sub<Sym(25)| (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), (1,13,21,18,9)(2,14,22,19,10)(3,15,23,20,6)(4,11,24,16,7)(5,12,25,17,8), (2,10,19,22,14)(3,20,15,6,23)(4,24,7,11,16)(5,12,25,17,8), (2,4,8,23)(3,19,16,25)(5,20,14,11)(6,22,24,12)(7,17,15,10)>;
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), (1,13,21,18,9)(2,14,22,19,10)(3,15,23,20,6)(4,11,24,16,7)(5,12,25,17,8), (2,10,19,22,14)(3,20,15,6,23)(4,24,7,11,16)(5,12,25,17,8), (2,4,8,23)(3,19,16,25)(5,20,14,11)(6,22,24,12)(7,17,15,10) );
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)], [(1,13,21,18,9),(2,14,22,19,10),(3,15,23,20,6),(4,11,24,16,7),(5,12,25,17,8)], [(2,10,19,22,14),(3,20,15,6,23),(4,24,7,11,16),(5,12,25,17,8)], [(2,4,8,23),(3,19,16,25),(5,20,14,11),(6,22,24,12),(7,17,15,10)]])
G:=TransitiveGroup(25,35);
Matrix representation of He5⋊4C4 ►in GL5(𝔽41)
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 37 |
| 18 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(41))| [0,0,0,0,18,1,0,0,0,0,0,16,0,0,0,0,0,10,0,0,0,0,0,37,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,10,0,0,0,0,0,37,0,0,0,0,0,18],[1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0] >;
He5⋊4C4 in GAP, Magma, Sage, TeX
{\rm He}_5\rtimes_4C_4 % in TeX
G:=Group("He5:4C4"); // GroupNames label
G:=SmallGroup(500,25);
// by ID
G=gap.SmallGroup(500,25);
# by ID
G:=PCGroup([5,-2,-2,-5,-5,-5,10,182,127,803,808,613]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a^2*b^-1*c,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of He5⋊4C4 in TeX
Character table of He5⋊4C4 in TeX