metabelian, supersoluble, monomial
Aliases: D20⋊3D5, Dic10⋊3D5, D10.2D10, C20.22D10, Dic5.1D10, C4.11D52, (C5×D20)⋊5C2, C5⋊D20⋊3C2, (D5×Dic5)⋊1C2, C52⋊3(C4○D4), C5⋊1(D4⋊2D5), C5⋊2(Q8⋊2D5), (C5×Dic10)⋊5C2, (C5×C10).3C23, C10.3(C22×D5), (C5×C20).18C22, (D5×C10).2C22, C52⋊6C4.9C22, (C5×Dic5).2C22, C2.6(C2×D52), (C4×C5⋊D5)⋊1C2, (C2×C5⋊D5).12C22, SmallGroup(400,165)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊D5
G = < a,b,c,d | a20=b2=c5=d2=1, bab=a-1, ac=ca, dad=a9, bc=cb, dbd=a18b, dcd=c-1 >
Subgroups: 604 in 96 conjugacy classes, 32 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C5×D5, C5⋊D5, C5×C10, D4⋊2D5, Q8⋊2D5, C5×Dic5, C52⋊6C4, C5×C20, D5×C10, C2×C5⋊D5, D5×Dic5, C5⋊D20, C5×Dic10, C5×D20, C4×C5⋊D5, D20⋊D5
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, D4⋊2D5, Q8⋊2D5, D52, C2×D52, D20⋊D5
►On 40 points
(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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 40)(20 39)
(1 17 13 9 5)(2 18 14 10 6)(3 19 15 11 7)(4 20 16 12 8)(21 25 29 33 37)(22 26 30 34 38)(23 27 31 35 39)(24 28 32 36 40)
(1 5)(2 14)(4 12)(6 10)(7 19)(9 17)(11 15)(16 20)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)
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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39), (1,17,13,9,5)(2,18,14,10,6)(3,19,15,11,7)(4,20,16,12,8)(21,25,29,33,37)(22,26,30,34,38)(23,27,31,35,39)(24,28,32,36,40), (1,5)(2,14)(4,12)(6,10)(7,19)(9,17)(11,15)(16,20)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)>;
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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39), (1,17,13,9,5)(2,18,14,10,6)(3,19,15,11,7)(4,20,16,12,8)(21,25,29,33,37)(22,26,30,34,38)(23,27,31,35,39)(24,28,32,36,40), (1,5)(2,14)(4,12)(6,10)(7,19)(9,17)(11,15)(16,20)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38) );
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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,40),(20,39)], [(1,17,13,9,5),(2,18,14,10,6),(3,19,15,11,7),(4,20,16,12,8),(21,25,29,33,37),(22,26,30,34,38),(23,27,31,35,39),(24,28,32,36,40)], [(1,5),(2,14),(4,12),(6,10),(7,19),(9,17),(11,15),(16,20),(21,23),(22,32),(24,30),(25,39),(26,28),(27,37),(29,35),(31,33),(34,40),(36,38)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 10 | 10 | 50 | 2 | 10 | 10 | 25 | 25 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 20 | 20 | 20 | 20 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D5 | C4○D4 | D10 | D10 | D10 | D4⋊2D5 | Q8⋊2D5 | D52 | C2×D52 | D20⋊D5 |
| kernel | D20⋊D5 | D5×Dic5 | C5⋊D20 | C5×Dic10 | C5×D20 | C4×C5⋊D5 | Dic10 | D20 | C52 | Dic5 | C20 | D10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of D20⋊D5 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,40,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,1,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,34,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40] >;
D20⋊D5 in GAP, Magma, Sage, TeX
D_{20}\rtimes D_5 % in TeX
G:=Group("D20:D5"); // GroupNames label
G:=SmallGroup(400,165);
// by ID
G=gap.SmallGroup(400,165);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,116,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^9,b*c=c*b,d*b*d=a^18*b,d*c*d=c^-1>;
// generators/relations