metabelian, supersoluble, monomial, A-group
Aliases: C5⋊2F5, C52⋊4C4, C5⋊D5.2C2, SmallGroup(100,12)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C4 |
Generators and relations for C52⋊C4
G = < a,b,c | a5=b5=c4=1, ab=ba, cac-1=a2, cbc-1=b3 >
Character table of C52⋊C4
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | |
| size | 1 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | orthogonal lifted from F5 |
| ρ6 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | orthogonal lifted from F5 |
| ρ7 | 4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | -1 | -1 | 3-√5/2 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1 | -1 | -1-√5 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | -1 | -1 | 3+√5/2 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1 | -1 | -1+√5 | orthogonal faithful |
►On 10 points - transitive group 10T10
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 3 5 2 4)(6 9 7 10 8)
(1 6)(2 9 5 8)(3 7 4 10)
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,3,5,2,4)(6,9,7,10,8), (1,6)(2,9,5,8)(3,7,4,10)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,3,5,2,4)(6,9,7,10,8), (1,6)(2,9,5,8)(3,7,4,10) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,3,5,2,4),(6,9,7,10,8)], [(1,6),(2,9,5,8),(3,7,4,10)]])
G:=TransitiveGroup(10,10);
►On 20 points - transitive group 20T27
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 18 9 12)(2 16 8 14)(3 19 7 11)(4 17 6 13)(5 20 10 15)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,18,9,12)(2,16,8,14)(3,19,7,11)(4,17,6,13)(5,20,10,15)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5,4,3,2)(6,10,9,8,7)(11,12,13,14,15)(16,17,18,19,20), (1,18,9,12)(2,16,8,14)(3,19,7,11)(4,17,6,13)(5,20,10,15) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5,4,3,2),(6,10,9,8,7),(11,12,13,14,15),(16,17,18,19,20)], [(1,18,9,12),(2,16,8,14),(3,19,7,11),(4,17,6,13),(5,20,10,15)]])
G:=TransitiveGroup(20,27);
►On 25 points - transitive group 25T10
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 25 19 14 9)(2 21 20 15 10)(3 22 16 11 6)(4 23 17 12 7)(5 24 18 13 8)
(2 4 5 3)(6 15 23 18)(7 13 22 20)(8 11 21 17)(9 14 25 19)(10 12 24 16)
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), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (2,4,5,3)(6,15,23,18)(7,13,22,20)(8,11,21,17)(9,14,25,19)(10,12,24,16)>;
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), (1,25,19,14,9)(2,21,20,15,10)(3,22,16,11,6)(4,23,17,12,7)(5,24,18,13,8), (2,4,5,3)(6,15,23,18)(7,13,22,20)(8,11,21,17)(9,14,25,19)(10,12,24,16) );
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)], [(1,25,19,14,9),(2,21,20,15,10),(3,22,16,11,6),(4,23,17,12,7),(5,24,18,13,8)], [(2,4,5,3),(6,15,23,18),(7,13,22,20),(8,11,21,17),(9,14,25,19),(10,12,24,16)]])
G:=TransitiveGroup(25,10);
C52⋊C4 is a maximal subgroup of
D5⋊F5 D5≀C2 C52⋊Q8 C52⋊Dic3 C15⋊2F5 He5⋊C4 C25⋊2F5 C53⋊6C4 C53⋊9C4
C52⋊C4 is a maximal quotient of
C52⋊5C8 C15⋊2F5 C25⋊2F5 He5⋊4C4 C53⋊6C4 C53⋊9C4
| action | f(x) | Disc(f) |
|---|---|---|
| 10T10 | x10-5x9-40x8+165x7+560x6-1611x5-2485x4+4870x3-620x2-1080x+144 | 224·326·515·112·194·712 |
Matrix representation of C52⋊C4 ►in GL4(𝔽41) generated by
| 35 | 40 | 0 | 0 |
| 36 | 40 | 0 | 0 |
| 1 | 0 | 40 | 34 |
| 34 | 34 | 7 | 7 |
| 40 | 1 | 0 | 0 |
| 5 | 35 | 0 | 0 |
| 6 | 1 | 40 | 34 |
| 39 | 35 | 7 | 7 |
| 0 | 0 | 40 | 1 |
| 7 | 1 | 39 | 34 |
| 0 | 0 | 40 | 0 |
| 6 | 1 | 40 | 0 |
G:=sub<GL(4,GF(41))| [35,36,1,34,40,40,0,34,0,0,40,7,0,0,34,7],[40,5,6,39,1,35,1,35,0,0,40,7,0,0,34,7],[0,7,0,6,0,1,0,1,40,39,40,40,1,34,0,0] >;
C52⋊C4 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_4
% in TeX
G:=Group("C5^2:C4"); // GroupNames label
G:=SmallGroup(100,12);
// by ID
G=gap.SmallGroup(100,12);
# by ID
G:=PCGroup([4,-2,-2,-5,-5,8,146,102,643,647]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C52⋊C4 in TeX
Character table of C52⋊C4 in TeX