metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊4D10, Q8⋊4D10, C20.49D4, C20.17C23, D20.11C22, D4⋊D5⋊6C2, C4○D4⋊1D5, Q8⋊D5⋊6C2, C5⋊5(C8⋊C22), (C2×C10).8D4, (C2×D20)⋊10C2, C5⋊2C8⋊4C22, C10.59(C2×D4), (C2×C4).22D10, (C5×D4)⋊4C22, C4.Dic5⋊9C2, (C5×Q8)⋊4C22, C4.24(C5⋊D4), C4.17(C22×D5), (C2×C20).42C22, C22.5(C5⋊D4), (C5×C4○D4)⋊1C2, C2.23(C2×C5⋊D4), SmallGroup(160,170)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D10
G = < a,b,c,d | a4=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a2b, dbd=a-1b, dcd=c-1 >
Subgroups: 256 in 68 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, M4(2), D8, SD16, C2×D4, C4○D4, C20, C20, D10, C2×C10, C2×C10, C8⋊C22, C5⋊2C8, D20, D20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C4.Dic5, D4⋊D5, Q8⋊D5, C2×D20, C5×C4○D4, D4⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C8⋊C22, C5⋊D4, C22×D5, C2×C5⋊D4, D4⋊D10
►On 40 points
(1 11 6 19)(2 12 7 20)(3 13 8 16)(4 14 9 17)(5 15 10 18)(21 32 26 37)(22 33 27 38)(23 34 28 39)(24 35 29 40)(25 36 30 31)
(1 38)(2 34)(3 40)(4 36)(5 32)(6 33)(7 39)(8 35)(9 31)(10 37)(11 27)(12 23)(13 29)(14 25)(15 21)(16 24)(17 30)(18 26)(19 22)(20 28)
(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 17)(2 16)(3 20)(4 19)(5 18)(6 14)(7 13)(8 12)(9 11)(10 15)(22 30)(23 29)(24 28)(25 27)(31 38)(32 37)(33 36)(34 35)(39 40)
G:=sub<Sym(40)| (1,11,6,19)(2,12,7,20)(3,13,8,16)(4,14,9,17)(5,15,10,18)(21,32,26,37)(22,33,27,38)(23,34,28,39)(24,35,29,40)(25,36,30,31), (1,38)(2,34)(3,40)(4,36)(5,32)(6,33)(7,39)(8,35)(9,31)(10,37)(11,27)(12,23)(13,29)(14,25)(15,21)(16,24)(17,30)(18,26)(19,22)(20,28), (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), (1,17)(2,16)(3,20)(4,19)(5,18)(6,14)(7,13)(8,12)(9,11)(10,15)(22,30)(23,29)(24,28)(25,27)(31,38)(32,37)(33,36)(34,35)(39,40)>;
G:=Group( (1,11,6,19)(2,12,7,20)(3,13,8,16)(4,14,9,17)(5,15,10,18)(21,32,26,37)(22,33,27,38)(23,34,28,39)(24,35,29,40)(25,36,30,31), (1,38)(2,34)(3,40)(4,36)(5,32)(6,33)(7,39)(8,35)(9,31)(10,37)(11,27)(12,23)(13,29)(14,25)(15,21)(16,24)(17,30)(18,26)(19,22)(20,28), (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), (1,17)(2,16)(3,20)(4,19)(5,18)(6,14)(7,13)(8,12)(9,11)(10,15)(22,30)(23,29)(24,28)(25,27)(31,38)(32,37)(33,36)(34,35)(39,40) );
G=PermutationGroup([[(1,11,6,19),(2,12,7,20),(3,13,8,16),(4,14,9,17),(5,15,10,18),(21,32,26,37),(22,33,27,38),(23,34,28,39),(24,35,29,40),(25,36,30,31)], [(1,38),(2,34),(3,40),(4,36),(5,32),(6,33),(7,39),(8,35),(9,31),(10,37),(11,27),(12,23),(13,29),(14,25),(15,21),(16,24),(17,30),(18,26),(19,22),(20,28)], [(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)], [(1,17),(2,16),(3,20),(4,19),(5,18),(6,14),(7,13),(8,12),(9,11),(10,15),(22,30),(23,29),(24,28),(25,27),(31,38),(32,37),(33,36),(34,35),(39,40)]])
D4⋊D10 is a maximal subgroup of
D4⋊4D20 D4.10D20 D4.3D20 D4.4D20 M4(2).D10 M4(2).15D10 2+ 1+4⋊D5 2- 1+4⋊2D5 D8⋊15D10 D8⋊11D10 D5×C8⋊C22 D40⋊C22 C20.C24 D20.32C23 D20.34C23 C60.36D4 C60.38D4 D60.C22 D12⋊D10 D4⋊D30 C20.3S4
D4⋊D10 is a maximal quotient of
C20.64(C4⋊C4) C4⋊C4⋊36D10 C4⋊C4.236D10 C20.38SD16 C42.48D10 C20⋊7D8 C20.23Q16 C42.56D10 Q8⋊D20 C4⋊D4.D5 D20⋊16D4 C4⋊D4⋊D5 (C2×C10).Q16 D20.36D4 C22⋊Q8⋊D5 C42.62D10 D20.23D4 C42.64D10 C42.68D10 D20.4Q8 C42.70D10 C4○D4⋊Dic5 (C5×D4)⋊14D4 C60.36D4 C60.38D4 D60.C22 D12⋊D10 D4⋊D30
31 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4⋊D10 |
| kernel | D4⋊D10 | C4.Dic5 | D4⋊D5 | Q8⋊D5 | C2×D20 | C5×C4○D4 | C20 | C2×C10 | C4○D4 | C2×C4 | D4 | Q8 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 4 |
Matrix representation of D4⋊D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 1 | 34 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40,0,0,0],[1,7,0,0,0,0,34,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D4⋊D10 in GAP, Magma, Sage, TeX
D_4\rtimes D_{10} % in TeX
G:=Group("D4:D10"); // GroupNames label
G:=SmallGroup(160,170);
// by ID
G=gap.SmallGroup(160,170);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,188,579,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations