non-abelian, soluble, monomial, A-group
Aliases: C52⋊Dic3, C5⋊D5.S3, C52⋊C3⋊1C4, C52⋊C6.2C2, SmallGroup(300,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C52⋊C3 — C52⋊Dic3 |
Generators and relations for C52⋊Dic3
G = < a,b,c,d | a5=b5=c6=1, d2=c3, ab=ba, cac-1=a-1b2, dad-1=a2b, cbc-1=ab2, dbd-1=b3, dcd-1=c-1 >
Character table of C52⋊Dic3
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | |
| size | 1 | 25 | 50 | 75 | 75 | 12 | 12 | 50 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | i | -i | 1 | 1 | -1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | -i | i | 1 | 1 | -1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 2 | 2 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 12 | 0 | 0 | 0 | 0 | 2 | -3 | 0 | orthogonal faithful |
| ρ8 | 12 | 0 | 0 | 0 | 0 | -3 | 2 | 0 | orthogonal faithful |
►On 15 points - transitive group 15T17
Generators in S15
(1 8 13 10 5)(2 9 14 11 6)
(1 8 13 10 5)(2 11 9 6 14)(3 15 7 4 12)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
(2 3)(4 11 7 14)(5 10 8 13)(6 15 9 12)
G:=sub<Sym(15)| (1,8,13,10,5)(2,9,14,11,6), (1,8,13,10,5)(2,11,9,6,14)(3,15,7,4,12), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (2,3)(4,11,7,14)(5,10,8,13)(6,15,9,12)>;
G:=Group( (1,8,13,10,5)(2,9,14,11,6), (1,8,13,10,5)(2,11,9,6,14)(3,15,7,4,12), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (2,3)(4,11,7,14)(5,10,8,13)(6,15,9,12) );
G=PermutationGroup([[(1,8,13,10,5),(2,9,14,11,6)], [(1,8,13,10,5),(2,11,9,6,14),(3,15,7,4,12)], [(1,2,3),(4,5,6,7,8,9),(10,11,12,13,14,15)], [(2,3),(4,11,7,14),(5,10,8,13),(6,15,9,12)]])
G:=TransitiveGroup(15,17);
►On 25 points: primitive - transitive group 25T28
Generators in S25
(1 21 13 10 24)(2 6 9 16 8)(3 5 11 19 12)(4 17 23 22 15)(7 18 25 20 14)
(1 2 14 17 5)(3 24 8 20 4)(6 7 23 11 21)(9 18 22 19 13)(10 16 25 15 12)
(2 3 4 5 6 7)(8 9 10 11 12 13)(14 15 16 17 18 19)(20 21 22 23 24 25)
(2 14 5 17)(3 19 6 16)(4 18 7 15)(8 22 11 25)(9 21 12 24)(10 20 13 23)
G:=sub<Sym(25)| (1,21,13,10,24)(2,6,9,16,8)(3,5,11,19,12)(4,17,23,22,15)(7,18,25,20,14), (1,2,14,17,5)(3,24,8,20,4)(6,7,23,11,21)(9,18,22,19,13)(10,16,25,15,12), (2,3,4,5,6,7)(8,9,10,11,12,13)(14,15,16,17,18,19)(20,21,22,23,24,25), (2,14,5,17)(3,19,6,16)(4,18,7,15)(8,22,11,25)(9,21,12,24)(10,20,13,23)>;
G:=Group( (1,21,13,10,24)(2,6,9,16,8)(3,5,11,19,12)(4,17,23,22,15)(7,18,25,20,14), (1,2,14,17,5)(3,24,8,20,4)(6,7,23,11,21)(9,18,22,19,13)(10,16,25,15,12), (2,3,4,5,6,7)(8,9,10,11,12,13)(14,15,16,17,18,19)(20,21,22,23,24,25), (2,14,5,17)(3,19,6,16)(4,18,7,15)(8,22,11,25)(9,21,12,24)(10,20,13,23) );
G=PermutationGroup([[(1,21,13,10,24),(2,6,9,16,8),(3,5,11,19,12),(4,17,23,22,15),(7,18,25,20,14)], [(1,2,14,17,5),(3,24,8,20,4),(6,7,23,11,21),(9,18,22,19,13),(10,16,25,15,12)], [(2,3,4,5,6,7),(8,9,10,11,12,13),(14,15,16,17,18,19),(20,21,22,23,24,25)], [(2,14,5,17),(3,19,6,16),(4,18,7,15),(8,22,11,25),(9,21,12,24),(10,20,13,23)]])
G:=TransitiveGroup(25,28);
►On 30 points - transitive group 30T71
Generators in S30
(2 19 25 28 22)(3 20 26 29 23)(4 11 18 15 8)(6 10 17 14 7)
(1 27 24 21 30)(2 19 25 28 22)(3 29 20 23 26)(4 15 11 8 18)(5 13 9 12 16)(6 10 17 14 7)
(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)
(1 4)(2 6)(3 5)(7 28 10 25)(8 27 11 30)(9 26 12 29)(13 20 16 23)(14 19 17 22)(15 24 18 21)
G:=sub<Sym(30)| (2,19,25,28,22)(3,20,26,29,23)(4,11,18,15,8)(6,10,17,14,7), (1,27,24,21,30)(2,19,25,28,22)(3,29,20,23,26)(4,15,11,8,18)(5,13,9,12,16)(6,10,17,14,7), (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), (1,4)(2,6)(3,5)(7,28,10,25)(8,27,11,30)(9,26,12,29)(13,20,16,23)(14,19,17,22)(15,24,18,21)>;
G:=Group( (2,19,25,28,22)(3,20,26,29,23)(4,11,18,15,8)(6,10,17,14,7), (1,27,24,21,30)(2,19,25,28,22)(3,29,20,23,26)(4,15,11,8,18)(5,13,9,12,16)(6,10,17,14,7), (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), (1,4)(2,6)(3,5)(7,28,10,25)(8,27,11,30)(9,26,12,29)(13,20,16,23)(14,19,17,22)(15,24,18,21) );
G=PermutationGroup([[(2,19,25,28,22),(3,20,26,29,23),(4,11,18,15,8),(6,10,17,14,7)], [(1,27,24,21,30),(2,19,25,28,22),(3,29,20,23,26),(4,15,11,8,18),(5,13,9,12,16),(6,10,17,14,7)], [(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)], [(1,4),(2,6),(3,5),(7,28,10,25),(8,27,11,30),(9,26,12,29),(13,20,16,23),(14,19,17,22),(15,24,18,21)]])
G:=TransitiveGroup(30,71);
Polynomial with Galois group C52⋊Dic3 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T17 | x15+45x13+630x11+162x10+3625x9+1350x8+9000x7+1800x6+7305x5-1375x3-2250x-450 | 222·322·540·116·134·1272·1949245823572 |
Matrix representation of C52⋊Dic3 ►in GL12(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(12,Integers())| [0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0],[0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0] >;
C52⋊Dic3 in GAP, Magma, Sage, TeX
C_5^2\rtimes {\rm Dic}_3 % in TeX
G:=Group("C5^2:Dic3"); // GroupNames label
G:=SmallGroup(300,23);
// by ID
G=gap.SmallGroup(300,23);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,5,10,122,5523,488,793,3004,3009,464]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^2*b,c*b*c^-1=a*b^2,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊Dic3 in TeX
Character table of C52⋊Dic3 in TeX