metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5D4, C42⋊6D10, Dic10⋊5D4, (C2×D4)⋊2D10, C4⋊1D4⋊4D5, C4.55(D4×D5), C5⋊3(D4⋊4D4), C20.35(C2×D4), D4⋊6D10⋊4C2, (C4×C20)⋊14C22, C20.D4⋊6C2, (D4×C10)⋊2C22, D20⋊4C4⋊13C2, C10.53C22≀C2, D4.D10⋊3C2, (C22×C10).23D4, C4.Dic5⋊7C22, (C2×C20).395C23, C4○D20.21C22, C23.11(C5⋊D4), C2.21(C23⋊D10), (C5×C4⋊1D4)⋊4C2, (C2×C10).526(C2×D4), C22.33(C2×C5⋊D4), (C2×C4).118(C22×D5), SmallGroup(320,704)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊5D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=a-1, ac=ca, dad=a11, cbc-1=a5b, dbd=a10b, dcd=c-1 >
Subgroups: 750 in 168 conjugacy classes, 39 normal (19 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×4], C22, C22 [×11], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×16], Q8 [×2], C23 [×2], C23 [×3], D5 [×2], C10, C10 [×4], C42, M4(2) [×2], D8 [×2], SD16 [×2], C2×D4 [×2], C2×D4 [×6], C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×7], C4.D4, C4≀C2 [×2], C4⋊1D4, C8⋊C22 [×2], 2+ 1+4, C5⋊2C8 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20, C2×C20, C5×D4 [×8], C22×D5 [×2], C22×C10 [×2], C22×C10, D4⋊4D4, C4.Dic5 [×2], D4⋊D5 [×2], D4.D5 [×2], C4×C20, C4○D20 [×2], D4×D5 [×2], D4⋊2D5 [×2], C2×C5⋊D4 [×2], D4×C10 [×2], D4×C10 [×2], D20⋊4C4 [×2], C20.D4, D4.D10 [×2], C5×C4⋊1D4, D4⋊6D10, D20⋊5D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C5⋊D4 [×2], C22×D5, D4⋊4D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, D20⋊5D4
►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 25)(2 24)(3 23)(4 22)(5 21)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)
(1 16 11 6)(2 17 12 7)(3 18 13 8)(4 19 14 9)(5 20 15 10)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 26)(22 37)(23 28)(24 39)(25 30)(27 32)(29 34)(31 36)(33 38)(35 40)
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,26)(22,37)(23,28)(24,39)(25,30)(27,32)(29,34)(31,36)(33,38)(35,40)>;
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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26), (1,16,11,6)(2,17,12,7)(3,18,13,8)(4,19,14,9)(5,20,15,10)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40), (1,6)(2,17)(3,8)(4,19)(5,10)(7,12)(9,14)(11,16)(13,18)(15,20)(21,26)(22,37)(23,28)(24,39)(25,30)(27,32)(29,34)(31,36)(33,38)(35,40) );
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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26)], [(1,16,11,6),(2,17,12,7),(3,18,13,8),(4,19,14,9),(5,20,15,10),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40)], [(1,6),(2,17),(3,8),(4,19),(5,10),(7,12),(9,14),(11,16),(13,18),(15,20),(21,26),(22,37),(23,28),(24,39),(25,30),(27,32),(29,34),(31,36),(33,38),(35,40)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 4 | 4 | 8 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 40 | 40 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | C5⋊D4 | D4⋊4D4 | D4×D5 | D20⋊5D4 |
| kernel | D20⋊5D4 | D20⋊4C4 | C20.D4 | D4.D10 | C5×C4⋊1D4 | D4⋊6D10 | Dic10 | D20 | C22×C10 | C4⋊1D4 | C42 | C2×D4 | C23 | C5 | C4 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of D20⋊5D4 ►in GL4(𝔽41) generated by
| 23 | 36 | 0 | 0 |
| 23 | 18 | 0 | 0 |
| 23 | 37 | 0 | 25 |
| 22 | 37 | 16 | 0 |
| 1 | 0 | 9 | 0 |
| 1 | 0 | 25 | 25 |
| 0 | 0 | 40 | 0 |
| 18 | 23 | 40 | 0 |
| 1 | 39 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 18 | 23 | 40 | 0 |
| 18 | 23 | 0 | 40 |
| 1 | 39 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 23 | 0 | 1 |
| 0 | 23 | 1 | 0 |
G:=sub<GL(4,GF(41))| [23,23,23,22,36,18,37,37,0,0,0,16,0,0,25,0],[1,1,0,18,0,0,0,23,9,25,40,40,0,25,0,0],[1,1,18,18,39,40,23,23,0,0,40,0,0,0,0,40],[1,0,0,0,39,40,23,23,0,0,0,1,0,0,1,0] >;
D20⋊5D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_5D_4 % in TeX
G:=Group("D20:5D4"); // GroupNames label
G:=SmallGroup(320,704);
// by ID
G=gap.SmallGroup(320,704);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,219,1123,570,297,136,1684,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^11,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations