non-abelian, soluble, monomial, A-group
Aliases: C52⋊D6, C5⋊D5⋊S3, C52⋊S3⋊C2, C52⋊C6⋊C2, C52⋊C3⋊C22, SmallGroup(300,25)
Series: Derived ►Chief ►Lower central ►Upper central
| C52⋊C3 — C52⋊D6 |
Generators and relations for C52⋊D6
G = < a,b,c,d | a5=b5=c6=d2=1, ab=ba, cac-1=dad=a2b3, cbc-1=a-1b-1, dbd=a-1b3, dcd=c-1 >
15C2
15C2
25C2
25C3
3C5
3C5
75C22
25C6
25S3
25S3
3D5
3D5
15D5
15C10
15D5
15C10
25D6
15D10
15D10
3D52
Character table of C52⋊D6
| class | 1 | 2A | 2B | 2C | 3 | 5A | 5B | 5C | 5D | 6 | 10A | 10B | 10C | 10D | |
| size | 1 | 15 | 15 | 25 | 50 | 6 | 6 | 6 | 6 | 50 | 30 | 30 | 30 | 30 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 0 | 0 | -2 | -1 | 2 | 2 | 2 | 2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 0 | 0 | 2 | -1 | 2 | 2 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ7 | 6 | 0 | 2 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | -1-√5/2 | -1+√5/2 | 0 | 0 | orthogonal faithful |
| ρ8 | 6 | 0 | -2 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | 1-√5/2 | 1+√5/2 | 0 | 0 | orthogonal faithful |
| ρ9 | 6 | 2 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | 0 | -1+√5/2 | -1-√5/2 | orthogonal faithful |
| ρ10 | 6 | 0 | -2 | 0 | 0 | -3+√5/2 | 1+√5 | 1-√5 | -3-√5/2 | 0 | 1+√5/2 | 1-√5/2 | 0 | 0 | orthogonal faithful |
| ρ11 | 6 | -2 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | 0 | 1+√5/2 | 1-√5/2 | orthogonal faithful |
| ρ12 | 6 | 0 | 2 | 0 | 0 | -3-√5/2 | 1-√5 | 1+√5 | -3+√5/2 | 0 | -1+√5/2 | -1-√5/2 | 0 | 0 | orthogonal faithful |
| ρ13 | 6 | 2 | 0 | 0 | 0 | 1-√5 | -3+√5/2 | -3-√5/2 | 1+√5 | 0 | 0 | 0 | -1-√5/2 | -1+√5/2 | orthogonal faithful |
| ρ14 | 6 | -2 | 0 | 0 | 0 | 1+√5 | -3-√5/2 | -3+√5/2 | 1-√5 | 0 | 0 | 0 | 1-√5/2 | 1+√5/2 | orthogonal faithful |
►On 15 points - transitive group 15T18
Generators in S15
(1 6 12 15 9)(2 13 4 7 10)(3 8 14 11 5)
(2 4 10 13 7)(3 5 11 14 8)
(1 2 3)(4 5 6 7 8 9)(10 11 12 13 14 15)
(1 2)(4 9)(5 8)(6 7)(10 15)(11 14)(12 13)
G:=sub<Sym(15)| (1,6,12,15,9)(2,13,4,7,10)(3,8,14,11,5), (2,4,10,13,7)(3,5,11,14,8), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (1,2)(4,9)(5,8)(6,7)(10,15)(11,14)(12,13)>;
G:=Group( (1,6,12,15,9)(2,13,4,7,10)(3,8,14,11,5), (2,4,10,13,7)(3,5,11,14,8), (1,2,3)(4,5,6,7,8,9)(10,11,12,13,14,15), (1,2)(4,9)(5,8)(6,7)(10,15)(11,14)(12,13) );
G=PermutationGroup([[(1,6,12,15,9),(2,13,4,7,10),(3,8,14,11,5)], [(2,4,10,13,7),(3,5,11,14,8)], [(1,2,3),(4,5,6,7,8,9),(10,11,12,13,14,15)], [(1,2),(4,9),(5,8),(6,7),(10,15),(11,14),(12,13)]])
G:=TransitiveGroup(15,18);
►On 25 points: primitive - transitive group 25T27
Generators in S25
(1 3 20 23 6)(2 19 13 17 7)(4 14 10 16 5)(8 12 22 15 21)(9 11 24 18 25)
(1 9 14 17 12)(2 15 20 24 16)(3 11 10 7 22)(4 13 8 6 25)(5 19 21 23 18)
(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 3)(4 7)(5 6)(9 13)(10 12)(14 18)(15 17)(20 25)(21 24)(22 23)
G:=sub<Sym(25)| (1,3,20,23,6)(2,19,13,17,7)(4,14,10,16,5)(8,12,22,15,21)(9,11,24,18,25), (1,9,14,17,12)(2,15,20,24,16)(3,11,10,7,22)(4,13,8,6,25)(5,19,21,23,18), (2,3,4,5,6,7)(8,9,10,11,12,13)(14,15,16,17,18,19)(20,21,22,23,24,25), (2,3)(4,7)(5,6)(9,13)(10,12)(14,18)(15,17)(20,25)(21,24)(22,23)>;
G:=Group( (1,3,20,23,6)(2,19,13,17,7)(4,14,10,16,5)(8,12,22,15,21)(9,11,24,18,25), (1,9,14,17,12)(2,15,20,24,16)(3,11,10,7,22)(4,13,8,6,25)(5,19,21,23,18), (2,3,4,5,6,7)(8,9,10,11,12,13)(14,15,16,17,18,19)(20,21,22,23,24,25), (2,3)(4,7)(5,6)(9,13)(10,12)(14,18)(15,17)(20,25)(21,24)(22,23) );
G=PermutationGroup([[(1,3,20,23,6),(2,19,13,17,7),(4,14,10,16,5),(8,12,22,15,21),(9,11,24,18,25)], [(1,9,14,17,12),(2,15,20,24,16),(3,11,10,7,22),(4,13,8,6,25),(5,19,21,23,18)], [(2,3,4,5,6,7),(8,9,10,11,12,13),(14,15,16,17,18,19),(20,21,22,23,24,25)], [(2,3),(4,7),(5,6),(9,13),(10,12),(14,18),(15,17),(20,25),(21,24),(22,23)]])
G:=TransitiveGroup(25,27);
►On 30 points - transitive group 30T66
Generators in S30
(2 9 26 15 23)(3 10 27 16 24)(5 20 18 29 12)(6 21 13 30 7)
(1 25 22 8 14)(2 15 9 23 26)(3 10 27 16 24)(4 17 11 19 28)(5 29 20 12 18)(6 21 13 30 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 3)(4 6)(7 11)(8 10)(13 17)(14 16)(19 21)(22 24)(25 27)(28 30)
G:=sub<Sym(30)| (2,9,26,15,23)(3,10,27,16,24)(5,20,18,29,12)(6,21,13,30,7), (1,25,22,8,14)(2,15,9,23,26)(3,10,27,16,24)(4,17,11,19,28)(5,29,20,12,18)(6,21,13,30,7), (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), (1,3)(4,6)(7,11)(8,10)(13,17)(14,16)(19,21)(22,24)(25,27)(28,30)>;
G:=Group( (2,9,26,15,23)(3,10,27,16,24)(5,20,18,29,12)(6,21,13,30,7), (1,25,22,8,14)(2,15,9,23,26)(3,10,27,16,24)(4,17,11,19,28)(5,29,20,12,18)(6,21,13,30,7), (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), (1,3)(4,6)(7,11)(8,10)(13,17)(14,16)(19,21)(22,24)(25,27)(28,30) );
G=PermutationGroup([[(2,9,26,15,23),(3,10,27,16,24),(5,20,18,29,12),(6,21,13,30,7)], [(1,25,22,8,14),(2,15,9,23,26),(3,10,27,16,24),(4,17,11,19,28),(5,29,20,12,18),(6,21,13,30,7)], [(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)], [(1,3),(4,6),(7,11),(8,10),(13,17),(14,16),(19,21),(22,24),(25,27),(28,30)]])
G:=TransitiveGroup(30,66);
►On 30 points - transitive group 30T72
Generators in S30
(2 7 24 26 14)(3 8 19 27 15)(5 17 29 21 10)(6 18 30 22 11)
(1 23 13 12 25)(2 26 7 14 24)(3 8 19 27 15)(4 28 9 16 20)(5 21 17 10 29)(6 18 30 22 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)
(1 6)(2 5)(3 4)(7 17)(8 16)(9 15)(10 14)(11 13)(12 18)(19 28)(20 27)(21 26)(22 25)(23 30)(24 29)
G:=sub<Sym(30)| (2,7,24,26,14)(3,8,19,27,15)(5,17,29,21,10)(6,18,30,22,11), (1,23,13,12,25)(2,26,7,14,24)(3,8,19,27,15)(4,28,9,16,20)(5,21,17,10,29)(6,18,30,22,11), (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), (1,6)(2,5)(3,4)(7,17)(8,16)(9,15)(10,14)(11,13)(12,18)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)>;
G:=Group( (2,7,24,26,14)(3,8,19,27,15)(5,17,29,21,10)(6,18,30,22,11), (1,23,13,12,25)(2,26,7,14,24)(3,8,19,27,15)(4,28,9,16,20)(5,21,17,10,29)(6,18,30,22,11), (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), (1,6)(2,5)(3,4)(7,17)(8,16)(9,15)(10,14)(11,13)(12,18)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29) );
G=PermutationGroup([[(2,7,24,26,14),(3,8,19,27,15),(5,17,29,21,10),(6,18,30,22,11)], [(1,23,13,12,25),(2,26,7,14,24),(3,8,19,27,15),(4,28,9,16,20),(5,21,17,10,29),(6,18,30,22,11)], [(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)], [(1,6),(2,5),(3,4),(7,17),(8,16),(9,15),(10,14),(11,13),(12,18),(19,28),(20,27),(21,26),(22,25),(23,30),(24,29)]])
G:=TransitiveGroup(30,72);
►On 30 points - transitive group 30T80
Generators in S30
(2 16 24 21 13)(3 17 19 22 14)(4 7 28 25 10)(6 12 27 30 9)
(1 23 18 15 20)(2 21 16 13 24)(3 17 19 22 14)(4 7 28 25 10)(5 26 8 11 29)(6 30 12 9 27)
(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 15)(8 14)(9 13)(10 18)(11 17)(12 16)(19 26)(20 25)(21 30)(22 29)(23 28)(24 27)
G:=sub<Sym(30)| (2,16,24,21,13)(3,17,19,22,14)(4,7,28,25,10)(6,12,27,30,9), (1,23,18,15,20)(2,21,16,13,24)(3,17,19,22,14)(4,7,28,25,10)(5,26,8,11,29)(6,30,12,9,27), (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)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)(19,26)(20,25)(21,30)(22,29)(23,28)(24,27)>;
G:=Group( (2,16,24,21,13)(3,17,19,22,14)(4,7,28,25,10)(6,12,27,30,9), (1,23,18,15,20)(2,21,16,13,24)(3,17,19,22,14)(4,7,28,25,10)(5,26,8,11,29)(6,30,12,9,27), (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)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)(19,26)(20,25)(21,30)(22,29)(23,28)(24,27) );
G=PermutationGroup([[(2,16,24,21,13),(3,17,19,22,14),(4,7,28,25,10),(6,12,27,30,9)], [(1,23,18,15,20),(2,21,16,13,24),(3,17,19,22,14),(4,7,28,25,10),(5,26,8,11,29),(6,30,12,9,27)], [(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),(7,15),(8,14),(9,13),(10,18),(11,17),(12,16),(19,26),(20,25),(21,30),(22,29),(23,28),(24,27)]])
G:=TransitiveGroup(30,80);
Polynomial with Galois group C52⋊D6 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 15T18 | x15-15x13-15x12+55x11+149x10+150x9-165x8-740x7-1105x6-1153x5-875x4-460x3-160x2-25x+1 | -518·172·235·532·1812·416517412 |
Matrix representation of C52⋊D6 ►in GL6(𝔽31)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 19 | 1 | 30 | 12 | 0 | 0 |
| 30 | 13 | 19 | 19 | 0 | 0 |
| 30 | 13 | 0 | 0 | 19 | 19 |
| 19 | 1 | 0 | 0 | 12 | 30 |
| 30 | 1 | 0 | 0 | 0 | 0 |
| 17 | 13 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 1 | 0 | 0 |
| 18 | 1 | 30 | 12 | 0 | 0 |
| 29 | 13 | 0 | 0 | 19 | 19 |
| 18 | 1 | 0 | 0 | 12 | 30 |
| 19 | 1 | 0 | 0 | 11 | 30 |
| 30 | 13 | 0 | 0 | 18 | 19 |
| 0 | 0 | 0 | 0 | 30 | 0 |
| 18 | 1 | 0 | 0 | 30 | 0 |
| 0 | 0 | 1 | 0 | 30 | 0 |
| 19 | 1 | 12 | 30 | 30 | 0 |
| 0 | 0 | 0 | 0 | 30 | 1 |
| 19 | 1 | 0 | 0 | 29 | 12 |
| 0 | 0 | 1 | 0 | 30 | 0 |
| 0 | 0 | 0 | 1 | 30 | 0 |
| 0 | 0 | 0 | 0 | 30 | 0 |
| 1 | 0 | 0 | 0 | 30 | 0 |
G:=sub<GL(6,GF(31))| [1,0,19,30,30,19,0,1,1,13,13,1,0,0,30,19,0,0,0,0,12,19,0,0,0,0,0,0,19,12,0,0,0,0,19,30],[30,17,30,18,29,18,1,13,0,1,13,1,0,0,0,30,0,0,0,0,1,12,0,0,0,0,0,0,19,12,0,0,0,0,19,30],[19,30,0,18,0,19,1,13,0,1,0,1,0,0,0,0,1,12,0,0,0,0,0,30,11,18,30,30,30,30,30,19,0,0,0,0],[0,19,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,30,29,30,30,30,30,1,12,0,0,0,0] >;
C52⋊D6 in GAP, Magma, Sage, TeX
C_5^2\rtimes D_6
% in TeX
G:=Group("C5^2:D6"); // GroupNames label
G:=SmallGroup(300,25);
// by ID
G=gap.SmallGroup(300,25);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,5,122,67,963,793,1804,3609,464]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^2*b^3,c*b*c^-1=a^-1*b^-1,d*b*d=a^-1*b^3,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C52⋊D6 in TeX
Character table of C52⋊D6 in TeX