non-abelian, soluble, monomial
Aliases: D52⋊1C4, C2.1D5≀C2, C5⋊D5.3D4, (C5×C10).1D4, C52⋊3(C22⋊C4), Dic5⋊2D5⋊4C2, (C2×D52).3C2, C5⋊D5.5(C2×C4), (C2×C52⋊C4)⋊2C2, (C2×C5⋊D5).6C22, SmallGroup(400,129)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D52⋊C4
G = < a,b,c,d,e | a5=b2=c5=d2=e4=1, bab=a-1, ac=ca, ad=da, eae-1=c, bc=cb, bd=db, ebe-1=d, dcd=c-1, ece-1=a, ede-1=b >
Subgroups: 630 in 66 conjugacy classes, 13 normal (11 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, C23, D5, C10, C22⋊C4, Dic5, C20, F5, D10, C2×C10, C52, C4×D5, C2×F5, C22×D5, C5×D5, C5⋊D5, C5×C10, C5×Dic5, C52⋊C4, D52, D52, D5×C10, C2×C5⋊D5, Dic5⋊2D5, C2×C52⋊C4, C2×D52, D52⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D5≀C2, D52⋊C4
Character table of D52⋊C4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 10 | 10 | 25 | 25 | 10 | 10 | 50 | 50 | 4 | 4 | 4 | 4 | 8 | 4 | 4 | 4 | 4 | 8 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -i | i | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | i | i | -i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | -i | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | -i | -i | i | linear of order 4 |
| ρ9 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | -4 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | 1+√5 | -3+√5/2 | 1-√5 | -3-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ12 | 4 | 4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | -1+√5 | 3+√5/2 | -1-√5 | 3-√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ13 | 4 | -4 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | 1-√5 | -3-√5/2 | 1+√5 | -3+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ14 | 4 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | -1-√5 | 3-√5/2 | -1+√5 | 3+√5/2 | -1 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ15 | 4 | 4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | -1-√5 | 3-√5/2 | -1+√5 | 3+√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ16 | 4 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | -1+√5 | 3+√5/2 | -1-√5 | 3-√5/2 | -1 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
| ρ17 | 4 | -4 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 3-√5/2 | 3+√5/2 | -1-√5 | -1+√5 | -1 | 1-√5 | -3-√5/2 | 1+√5 | -3+√5/2 | 1 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ18 | 4 | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | 3+√5/2 | -1-√5 | 3-√5/2 | -1+√5 | -1 | 0 | 0 | 0 | 0 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | 3+√5/2 | -1-√5 | 3-√5/2 | -1+√5 | -1 | 0 | 0 | 0 | 0 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ20 | 4 | 4 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | 3-√5/2 | -1+√5 | 3+√5/2 | -1-√5 | -1 | 0 | 0 | 0 | 0 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | orthogonal lifted from D5≀C2 |
| ρ21 | 4 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | 3-√5/2 | -1+√5 | 3+√5/2 | -1-√5 | -1 | 0 | 0 | 0 | 0 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | orthogonal lifted from D5≀C2 |
| ρ22 | 4 | -4 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 3+√5/2 | 3-√5/2 | -1+√5 | -1-√5 | -1 | 1+√5 | -3+√5/2 | 1-√5 | -3-√5/2 | 1 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | -3+√5/2 | 1-√5 | -3-√5/2 | 1+√5 | 1 | 0 | 0 | 0 | 0 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | -3-√5/2 | 1+√5 | -3+√5/2 | 1-√5 | 1 | 0 | 0 | 0 | 0 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -1-√5 | -1+√5 | 3+√5/2 | 3-√5/2 | -1 | -3+√5/2 | 1-√5 | -3-√5/2 | 1+√5 | 1 | 0 | 0 | 0 | 0 | ζ43ζ53+ζ43ζ52 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ54+ζ43ζ5 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | -1+√5 | -1-√5 | 3-√5/2 | 3+√5/2 | -1 | -3-√5/2 | 1+√5 | -3+√5/2 | 1-√5 | 1 | 0 | 0 | 0 | 0 | ζ43ζ54+ζ43ζ5 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ53+ζ43ζ52 | complex faithful |
| ρ27 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | 2 | 2 | 2 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ28 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 3 | -2 | -2 | -2 | -2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D5≀C2 |
►On 20 points - transitive group 20T93
Generators in S20
(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
G:=sub<Sym(20)| (11,12,13,14,15)(16,17,18,19,20), (1,6)(2,7)(3,8)(4,9)(5,10)(11,20)(12,19)(13,18)(14,17)(15,16), (1,2,3,4,5)(6,7,8,9,10), (1,10)(2,9)(3,8)(4,7)(5,6)(11,16)(12,17)(13,18)(14,19)(15,20), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)>;
G:=Group( (11,12,13,14,15)(16,17,18,19,20), (1,6)(2,7)(3,8)(4,9)(5,10)(11,20)(12,19)(13,18)(14,17)(15,16), (1,2,3,4,5)(6,7,8,9,10), (1,10)(2,9)(3,8)(4,7)(5,6)(11,16)(12,17)(13,18)(14,19)(15,20), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15) );
G=PermutationGroup([[(11,12,13,14,15),(16,17,18,19,20)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,20),(12,19),(13,18),(14,17),(15,16)], [(1,2,3,4,5),(6,7,8,9,10)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15)]])
G:=TransitiveGroup(20,93);
►On 20 points - transitive group 20T94
Generators in S20
(11 12 13 14 15)(16 17 18 19 20)
(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 7 8 9 10)
(1 10)(2 9)(3 8)(4 7)(5 6)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
G:=sub<Sym(20)| (11,12,13,14,15)(16,17,18,19,20), (11,20)(12,19)(13,18)(14,17)(15,16), (1,2,3,4,5)(6,7,8,9,10), (1,10)(2,9)(3,8)(4,7)(5,6), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)>;
G:=Group( (11,12,13,14,15)(16,17,18,19,20), (11,20)(12,19)(13,18)(14,17)(15,16), (1,2,3,4,5)(6,7,8,9,10), (1,10)(2,9)(3,8)(4,7)(5,6), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15) );
G=PermutationGroup([[(11,12,13,14,15),(16,17,18,19,20)], [(11,20),(12,19),(13,18),(14,17),(15,16)], [(1,2,3,4,5),(6,7,8,9,10)], [(1,10),(2,9),(3,8),(4,7),(5,6)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15)]])
G:=TransitiveGroup(20,94);
►On 20 points - transitive group 20T95
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 15)(2 14)(3 13)(4 12)(5 11)(6 16)(7 20)(8 19)(9 18)(10 17)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 18)(2 19)(3 20)(4 16)(5 17)(6 12)(7 13)(8 14)(9 15)(10 11)
(1 16 7 11)(2 19 8 14)(3 17 9 12)(4 20 10 15)(5 18 6 13)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,16)(7,20)(8,19)(9,18)(10,17), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,18)(2,19)(3,20)(4,16)(5,17)(6,12)(7,13)(8,14)(9,15)(10,11), (1,16,7,11)(2,19,8,14)(3,17,9,12)(4,20,10,15)(5,18,6,13)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,15)(2,14)(3,13)(4,12)(5,11)(6,16)(7,20)(8,19)(9,18)(10,17), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,18)(2,19)(3,20)(4,16)(5,17)(6,12)(7,13)(8,14)(9,15)(10,11), (1,16,7,11)(2,19,8,14)(3,17,9,12)(4,20,10,15)(5,18,6,13) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,16),(7,20),(8,19),(9,18),(10,17)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,18),(2,19),(3,20),(4,16),(5,17),(6,12),(7,13),(8,14),(9,15),(10,11)], [(1,16,7,11),(2,19,8,14),(3,17,9,12),(4,20,10,15),(5,18,6,13)]])
G:=TransitiveGroup(20,95);
Matrix representation of D52⋊C4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 35 | 5 | 6 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 6 | 34 | 0 |
| 0 | 0 | 0 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 34 | 40 | 35 | 35 |
| 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,1,0,35,0,0,0,0,1,5,0,0,1,0,0,6],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,1,34,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,40,40,1,0,0,0,0,0,1],[40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,34,40,1,0,0,0,40,0,0,0,0,0,35,0,1,0,0,0,35,1,0] >;
D52⋊C4 in GAP, Magma, Sage, TeX
D_5^2\rtimes C_4
% in TeX
G:=Group("D5^2:C4"); // GroupNames label
G:=SmallGroup(400,129);
// by ID
G=gap.SmallGroup(400,129);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,5,73,55,7204,1210,496,1157,299,2897]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^5=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=c,b*c=c*b,b*d=d*b,e*b*e^-1=d,d*c*d=c^-1,e*c*e^-1=a,e*d*e^-1=b>;
// generators/relations
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