non-abelian, soluble, monomial
Aliases: C52⋊M4(2), D52.C4, C52⋊C8⋊C2, C52⋊C4.C4, D5⋊F5.2C2, C5⋊F5.1C22, C5⋊D5.2(C2×C4), SmallGroup(400,206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C52⋊M4(2)
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, cac-1=b2, dad=a-1, cbc-1=a, bd=db, dcd=c5 >
10C2
25C2
2C5
4C5
25C22
25C4
25C4
2D5
10C10
10D5
20D5
25C8
25C2×C4
25C8
10F5
10F5
10D10
20F5
25M4(2)
10C2×F5
Character table of C52⋊M4(2)
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 8C | 8D | 10 | |
| size | 1 | 10 | 25 | 25 | 25 | 50 | 8 | 16 | 50 | 50 | 50 | 50 | 40 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -1 | linear of order 4 |
| ρ9 | 2 | 0 | -2 | 2i | -2i | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | 0 | -2 | -2i | 2i | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ11 | 8 | 4 | 0 | 0 | 0 | 0 | 3 | -2 | 0 | 0 | 0 | 0 | -1 | orthogonal faithful |
| ρ12 | 8 | -4 | 0 | 0 | 0 | 0 | 3 | -2 | 0 | 0 | 0 | 0 | 1 | orthogonal faithful |
| ρ13 | 16 | 0 | 0 | 0 | 0 | 0 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 10 points - transitive group 10T28
Generators in S10
(1 8 6 10 4)
(2 9 7 3 5)
(1 2)(3 4 5 6 7 8 9 10)
(4 8)(6 10)
G:=sub<Sym(10)| (1,8,6,10,4), (2,9,7,3,5), (1,2)(3,4,5,6,7,8,9,10), (4,8)(6,10)>;
G:=Group( (1,8,6,10,4), (2,9,7,3,5), (1,2)(3,4,5,6,7,8,9,10), (4,8)(6,10) );
G=PermutationGroup([[(1,8,6,10,4)], [(2,9,7,3,5)], [(1,2),(3,4,5,6,7,8,9,10)], [(4,8),(6,10)]])
G:=TransitiveGroup(10,28);
►On 20 points - transitive group 20T104
Generators in S20
(1 5 19 15 9)(3 13 11 7 17)
(2 6 20 16 10)(4 14 12 8 18)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(1 3)(2 4)(5 17)(6 14)(7 19)(8 16)(9 13)(10 18)(11 15)(12 20)
G:=sub<Sym(20)| (1,5,19,15,9)(3,13,11,7,17), (2,6,20,16,10)(4,14,12,8,18), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (1,3)(2,4)(5,17)(6,14)(7,19)(8,16)(9,13)(10,18)(11,15)(12,20)>;
G:=Group( (1,5,19,15,9)(3,13,11,7,17), (2,6,20,16,10)(4,14,12,8,18), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (1,3)(2,4)(5,17)(6,14)(7,19)(8,16)(9,13)(10,18)(11,15)(12,20) );
G=PermutationGroup([[(1,5,19,15,9),(3,13,11,7,17)], [(2,6,20,16,10),(4,14,12,8,18)], [(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)], [(1,3),(2,4),(5,17),(6,14),(7,19),(8,16),(9,13),(10,18),(11,15),(12,20)]])
G:=TransitiveGroup(20,104);
►On 20 points - transitive group 20T107
Generators in S20
(1 12 10 6 8)(2 9 7 11 5)(3 19 17 13 15)(4 18 16 20 14)
(1 10 8 12 6)(2 5 11 7 9)(3 19 17 13 15)(4 20 18 14 16)
(1 2)(3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(1 4)(2 3)(5 19)(6 16)(7 13)(8 18)(9 15)(10 20)(11 17)(12 14)
G:=sub<Sym(20)| (1,12,10,6,8)(2,9,7,11,5)(3,19,17,13,15)(4,18,16,20,14), (1,10,8,12,6)(2,5,11,7,9)(3,19,17,13,15)(4,20,18,14,16), (1,2)(3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (1,4)(2,3)(5,19)(6,16)(7,13)(8,18)(9,15)(10,20)(11,17)(12,14)>;
G:=Group( (1,12,10,6,8)(2,9,7,11,5)(3,19,17,13,15)(4,18,16,20,14), (1,10,8,12,6)(2,5,11,7,9)(3,19,17,13,15)(4,20,18,14,16), (1,2)(3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (1,4)(2,3)(5,19)(6,16)(7,13)(8,18)(9,15)(10,20)(11,17)(12,14) );
G=PermutationGroup([[(1,12,10,6,8),(2,9,7,11,5),(3,19,17,13,15),(4,18,16,20,14)], [(1,10,8,12,6),(2,5,11,7,9),(3,19,17,13,15),(4,20,18,14,16)], [(1,2),(3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)], [(1,4),(2,3),(5,19),(6,16),(7,13),(8,18),(9,15),(10,20),(11,17),(12,14)]])
G:=TransitiveGroup(20,107);
►On 20 points - transitive group 20T109
Generators in S20
(2 12 20 16 8)(4 14 10 6 18)
(1 15 11 7 19)(3 5 13 17 9)
(1 2 3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(6 10)(8 12)(14 18)(16 20)
G:=sub<Sym(20)| (2,12,20,16,8)(4,14,10,6,18), (1,15,11,7,19)(3,5,13,17,9), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (6,10)(8,12)(14,18)(16,20)>;
G:=Group( (2,12,20,16,8)(4,14,10,6,18), (1,15,11,7,19)(3,5,13,17,9), (1,2,3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (6,10)(8,12)(14,18)(16,20) );
G=PermutationGroup([[(2,12,20,16,8),(4,14,10,6,18)], [(1,15,11,7,19),(3,5,13,17,9)], [(1,2,3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)], [(6,10),(8,12),(14,18),(16,20)]])
G:=TransitiveGroup(20,109);
►On 20 points - transitive group 20T115
Generators in S20
(1 19 17 13 15)(3 12 10 6 8)
(2 20 18 14 16)(4 5 11 7 9)
(1 2)(3 4)(5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20)
(1 3)(2 4)(5 20)(6 17)(7 14)(8 19)(9 16)(10 13)(11 18)(12 15)
G:=sub<Sym(20)| (1,19,17,13,15)(3,12,10,6,8), (2,20,18,14,16)(4,5,11,7,9), (1,2)(3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (1,3)(2,4)(5,20)(6,17)(7,14)(8,19)(9,16)(10,13)(11,18)(12,15)>;
G:=Group( (1,19,17,13,15)(3,12,10,6,8), (2,20,18,14,16)(4,5,11,7,9), (1,2)(3,4)(5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20), (1,3)(2,4)(5,20)(6,17)(7,14)(8,19)(9,16)(10,13)(11,18)(12,15) );
G=PermutationGroup([[(1,19,17,13,15),(3,12,10,6,8)], [(2,20,18,14,16),(4,5,11,7,9)], [(1,2),(3,4),(5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20)], [(1,3),(2,4),(5,20),(6,17),(7,14),(8,19),(9,16),(10,13),(11,18),(12,15)]])
G:=TransitiveGroup(20,115);
►On 25 points: primitive - transitive group 25T31
Generators in S25
(1 5 3 7 9)(2 15 18 12 25)(4 20 19 17 14)(6 21 16 22 11)(8 10 13 23 24)
(1 6 4 8 2)(3 16 19 13 18)(5 21 20 10 15)(7 22 17 23 12)(9 11 14 24 25)
(2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17)(18 19 20 21 22 23 24 25)
(3 7)(5 9)(10 24)(11 21)(12 18)(13 23)(14 20)(15 25)(16 22)(17 19)
G:=sub<Sym(25)| (1,5,3,7,9)(2,15,18,12,25)(4,20,19,17,14)(6,21,16,22,11)(8,10,13,23,24), (1,6,4,8,2)(3,16,19,13,18)(5,21,20,10,15)(7,22,17,23,12)(9,11,14,24,25), (2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25), (3,7)(5,9)(10,24)(11,21)(12,18)(13,23)(14,20)(15,25)(16,22)(17,19)>;
G:=Group( (1,5,3,7,9)(2,15,18,12,25)(4,20,19,17,14)(6,21,16,22,11)(8,10,13,23,24), (1,6,4,8,2)(3,16,19,13,18)(5,21,20,10,15)(7,22,17,23,12)(9,11,14,24,25), (2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17)(18,19,20,21,22,23,24,25), (3,7)(5,9)(10,24)(11,21)(12,18)(13,23)(14,20)(15,25)(16,22)(17,19) );
G=PermutationGroup([[(1,5,3,7,9),(2,15,18,12,25),(4,20,19,17,14),(6,21,16,22,11),(8,10,13,23,24)], [(1,6,4,8,2),(3,16,19,13,18),(5,21,20,10,15),(7,22,17,23,12),(9,11,14,24,25)], [(2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17),(18,19,20,21,22,23,24,25)], [(3,7),(5,9),(10,24),(11,21),(12,18),(13,23),(14,20),(15,25),(16,22),(17,19)]])
G:=TransitiveGroup(25,31);
Polynomial with Galois group C52⋊M4(2) over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 10T28 | x10-6x9-45x8+136x7+1043x6+390x5-8071x4-22200x3-25799x2-14082x-2871 | 222·32·58·232·432·794 |
Matrix representation of C52⋊M4(2) ►in GL8(ℤ)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,1,0] >;
C52⋊M4(2) in GAP, Magma, Sage, TeX
C_5^2\rtimes M_4(2)
% in TeX
G:=Group("C5^2:M4(2)"); // GroupNames label
G:=SmallGroup(400,206);
// by ID
G=gap.SmallGroup(400,206);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,5,48,121,31,50,964,1210,256,262,8645,587,1457,1463]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^5=c^8=d^2=1,a*b=b*a,c*a*c^-1=b^2,d*a*d=a^-1,c*b*c^-1=a,b*d=d*b,d*c*d=c^5>;
// generators/relations
Export
Subgroup lattice of C52⋊M4(2) in TeX
Character table of C52⋊M4(2) in TeX