metabelian, soluble, monomial, A-group
Aliases: C52⋊C3, SmallGroup(75,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C52⋊C3 |
Generators and relations for C52⋊C3
G = < a,b,c | a5=b5=c3=1, ab=ba, cac-1=a3b3, cbc-1=a-1b >
Character table of C52⋊C3
| class | 1 | 3A | 3B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | |
| size | 1 | 25 | 25 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 3 | 0 | 0 | 1+√5/2 | 1-√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | complex faithful |
| ρ5 | 3 | 0 | 0 | ζ53+2ζ5 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | complex faithful |
| ρ6 | 3 | 0 | 0 | 2ζ52+ζ5 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 2ζ54+ζ52 | ζ54+2ζ53 | complex faithful |
| ρ7 | 3 | 0 | 0 | 1+√5/2 | 1-√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | 1-√5/2 | 1+√5/2 | complex faithful |
| ρ8 | 3 | 0 | 0 | 1-√5/2 | 1+√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | 2ζ52+ζ5 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | complex faithful |
| ρ9 | 3 | 0 | 0 | 2ζ54+ζ52 | 2ζ52+ζ5 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | ζ54+2ζ53 | ζ53+2ζ5 | complex faithful |
| ρ10 | 3 | 0 | 0 | ζ54+2ζ53 | 2ζ54+ζ52 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | ζ53+2ζ5 | 2ζ52+ζ5 | complex faithful |
| ρ11 | 3 | 0 | 0 | 1-√5/2 | 1+√5/2 | 2ζ52+ζ5 | 2ζ54+ζ52 | ζ54+2ζ53 | ζ53+2ζ5 | 1+√5/2 | 1-√5/2 | complex faithful |
►On 15 points - transitive group 15T9
Generators in S15
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)
(6 7 8 9 10)(11 14 12 15 13)
(1 13 9)(2 15 10)(3 12 6)(4 14 7)(5 11 8)
G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (6,7,8,9,10)(11,14,12,15,13), (1,13,9)(2,15,10)(3,12,6)(4,14,7)(5,11,8)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15), (6,7,8,9,10)(11,14,12,15,13), (1,13,9)(2,15,10)(3,12,6)(4,14,7)(5,11,8) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)], [(6,7,8,9,10),(11,14,12,15,13)], [(1,13,9),(2,15,10),(3,12,6),(4,14,7),(5,11,8)]])
G:=TransitiveGroup(15,9);
►On 25 points: primitive - transitive group 25T6
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 20 12)(3 6 21)(4 23 8)(5 13 16)(7 11 14)(9 18 10)(15 24 25)(17 19 22)
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), (2,20,12)(3,6,21)(4,23,8)(5,13,16)(7,11,14)(9,18,10)(15,24,25)(17,19,22)>;
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), (2,20,12)(3,6,21)(4,23,8)(5,13,16)(7,11,14)(9,18,10)(15,24,25)(17,19,22) );
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)], [(2,20,12),(3,6,21),(4,23,8),(5,13,16),(7,11,14),(9,18,10),(15,24,25),(17,19,22)]])
G:=TransitiveGroup(25,6);
C52⋊C3 is a maximal subgroup of
C52⋊S3 C52⋊C6 C52⋊A4
C52⋊C3 is a maximal quotient of C52⋊C9 C52⋊A4 He5⋊C3
| action | f(x) | Disc(f) |
|---|---|---|
| 15T9 | x15-470x13-305x12+71840x11+85357x10-4292700x9-3714805x8+119761820x7+25284495x6-1542190154x5+717324725x4+7178878600x3-5452953875x2-7998223215x+4461221029 | 224·36·524·750·4118·2932·15672·29272·16079412 |
Matrix representation of C52⋊C3 ►in GL3(𝔽11) generated by
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 5 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 9 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(11))| [1,0,0,0,9,0,0,0,5],[4,0,0,0,4,0,0,0,9],[0,1,0,0,0,1,1,0,0] >;
C52⋊C3 in GAP, Magma, Sage, TeX
C_5^2\rtimes C_3
% in TeX
G:=Group("C5^2:C3"); // GroupNames label
G:=SmallGroup(75,2);
// by ID
G=gap.SmallGroup(75,2);
# by ID
G:=PCGroup([3,-3,-5,5,199,434]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^3=1,a*b=b*a,c*a*c^-1=a^3*b^3,c*b*c^-1=a^-1*b>;
// generators/relations
Export
Subgroup lattice of C52⋊C3 in TeX
Character table of C52⋊C3 in TeX