metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.1D4, (C2×C4)⋊F5, (C2×C20)⋊1C4, C5⋊1(C23⋊C4), C22⋊F5⋊1C2, (C2×D20).2C2, (C22×D5)⋊2C4, C22.2(C2×F5), C2.4(C22⋊F5), C10.1(C22⋊C4), (C22×D5).14C22, (C2×C10).2(C2×C4), SmallGroup(160,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.D4
G = < a,b,c,d | a10=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a3, cbc-1=a7b, dbd-1=a2b, dcd-1=a4bc-1 >
2C2
10C2
10C2
20C2
2C4
5C22
5C22
10C22
20C4
20C4
20C22
20C22
2D5
2D5
2C10
4D5
5C23
5C23
10C2×C4
10D4
10D4
10C2×C4
2C20
2D10
4D10
4F5
4F5
4D10
2D20
2D20
Character table of D10.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 5 | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 2 | 10 | 10 | 20 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 4 | 4 | 4 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
| ρ13 | 4 | 4 | 4 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | √5 | √5 | -√5 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ15 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | -√5 | -√5 | √5 | √5 | orthogonal lifted from C22⋊F5 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √5 | -√5 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ43ζ52+2ζ43ζ5+ζ43 | orthogonal faithful |
| ρ17 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√5 | √5 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ4ζ53+2ζ4ζ5+ζ4 | orthogonal faithful |
| ρ18 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | -√5 | √5 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ53+ζ43 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ4ζ54+2ζ4ζ52+ζ4 | orthogonal faithful |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | √5 | -√5 | 2ζ4ζ54+2ζ4ζ52+ζ4 | 2ζ4ζ53+2ζ4ζ5+ζ4 | 2ζ43ζ52+2ζ43ζ5+ζ43 | 2ζ43ζ54+2ζ43ζ53+ζ43 | orthogonal faithful |
►On 40 points
Generators in S40
(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)(31 32 33 34 35 36 37 38 39 40)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)(21 38)(22 37)(23 36)(24 35)(25 34)(26 33)(27 32)(28 31)(29 40)(30 39)
(1 24 12 36)(2 21 11 39)(3 28 20 32)(4 25 19 35)(5 22 18 38)(6 29 17 31)(7 26 16 34)(8 23 15 37)(9 30 14 40)(10 27 13 33)
(1 31 17 29)(2 38 16 22)(3 35 15 25)(4 32 14 28)(5 39 13 21)(6 36 12 24)(7 33 11 27)(8 40 20 30)(9 37 19 23)(10 34 18 26)
G:=sub<Sym(40)| (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)(31,32,33,34,35,36,37,38,39,40), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,20)(8,19)(9,18)(10,17)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39), (1,24,12,36)(2,21,11,39)(3,28,20,32)(4,25,19,35)(5,22,18,38)(6,29,17,31)(7,26,16,34)(8,23,15,37)(9,30,14,40)(10,27,13,33), (1,31,17,29)(2,38,16,22)(3,35,15,25)(4,32,14,28)(5,39,13,21)(6,36,12,24)(7,33,11,27)(8,40,20,30)(9,37,19,23)(10,34,18,26)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,20)(8,19)(9,18)(10,17)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,40)(30,39), (1,24,12,36)(2,21,11,39)(3,28,20,32)(4,25,19,35)(5,22,18,38)(6,29,17,31)(7,26,16,34)(8,23,15,37)(9,30,14,40)(10,27,13,33), (1,31,17,29)(2,38,16,22)(3,35,15,25)(4,32,14,28)(5,39,13,21)(6,36,12,24)(7,33,11,27)(8,40,20,30)(9,37,19,23)(10,34,18,26) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,20),(8,19),(9,18),(10,17),(21,38),(22,37),(23,36),(24,35),(25,34),(26,33),(27,32),(28,31),(29,40),(30,39)], [(1,24,12,36),(2,21,11,39),(3,28,20,32),(4,25,19,35),(5,22,18,38),(6,29,17,31),(7,26,16,34),(8,23,15,37),(9,30,14,40),(10,27,13,33)], [(1,31,17,29),(2,38,16,22),(3,35,15,25),(4,32,14,28),(5,39,13,21),(6,36,12,24),(7,33,11,27),(8,40,20,30),(9,37,19,23),(10,34,18,26)]])
D10.D4 is a maximal subgroup of
C42⋊F5 C42⋊2F5 (C2×D4)⋊F5 (C2×Q8)⋊F5 C23⋊F5⋊5C2 (C2×D4)⋊7F5 (C2×Q8)⋊7F5 D10.4D12 (C2×C60)⋊C4
D10.D4 is a maximal quotient of
C42⋊F5 C42⋊2F5 C42.F5 C42.2F5 C5⋊C2≀C4 C22⋊C4⋊F5 D10.1D8 D10.1Q16 (C2×C20)⋊1C8 C5⋊(C23⋊C8) C22⋊F5⋊C4 D10.4D12 (C2×C60)⋊C4
Matrix representation of D10.D4 ►in GL4(𝔽41) generated by
| 6 | 6 | 0 | 0 |
| 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 27 | 27 | 1 | 35 |
| 25 | 16 | 0 | 0 |
| 2 | 16 | 0 | 0 |
| 32 | 27 | 28 | 39 |
| 14 | 30 | 2 | 13 |
| 40 | 13 | 37 | 15 |
| 7 | 39 | 26 | 4 |
| 4 | 16 | 14 | 9 |
| 6 | 5 | 11 | 30 |
| 34 | 0 | 39 | 0 |
| 1 | 1 | 0 | 39 |
| 5 | 27 | 7 | 0 |
| 30 | 20 | 40 | 40 |
G:=sub<GL(4,GF(41))| [6,35,0,27,6,1,0,27,0,0,0,1,0,0,40,35],[25,2,32,14,16,16,27,30,0,0,28,2,0,0,39,13],[40,7,4,6,13,39,16,5,37,26,14,11,15,4,9,30],[34,1,5,30,0,1,27,20,39,0,7,40,0,39,0,40] >;
D10.D4 in GAP, Magma, Sage, TeX
D_{10}.D_4 % in TeX
G:=Group("D10.D4"); // GroupNames label
G:=SmallGroup(160,74);
// by ID
G=gap.SmallGroup(160,74);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,103,188,579,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^3,c*b*c^-1=a^7*b,d*b*d^-1=a^2*b,d*c*d^-1=a^4*b*c^-1>;
// generators/relations
Export
Subgroup lattice of D10.D4 in TeX
Character table of D10.D4 in TeX