metabelian, supersoluble, monomial
Aliases: D10.9D10, C20.32D10, Dic5.10D10, C4.17D52, (C4×D5)⋊4D5, (D5×C20)⋊1C2, C5⋊2(C4○D20), C5⋊D20⋊7C2, C52⋊4(C4○D4), C52⋊2D4⋊6C2, C52⋊2Q8⋊6C2, (C5×C10).5C23, C10.5(C22×D5), (C5×C20).31C22, (D5×C10).10C22, C52⋊6C4.11C22, (C5×Dic5).14C22, C2.8(C2×D52), (C4×C5⋊D5)⋊6C2, (C2×C5⋊D5).14C22, SmallGroup(400,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.9D10
G = < a,b,c,d | a10=b2=1, c10=d2=a5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c9 >
Subgroups: 604 in 96 conjugacy classes, 32 normal (12 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, C4×D5, D20, C5⋊D4, C2×C20, C5×D5, C5⋊D5, C5×C10, C4○D20, C5×Dic5, C52⋊6C4, C5×C20, D5×C10, C2×C5⋊D5, C52⋊2D4, C5⋊D20, C52⋊2Q8, D5×C20, C4×C5⋊D5, D10.9D10
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C4○D20, D52, C2×D52, D10.9D10
►On 40 points
(1 19 17 15 13 11 9 7 5 3)(2 20 18 16 14 12 10 8 6 4)(21 23 25 27 29 31 33 35 37 39)(22 24 26 28 30 32 34 36 38 40)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 21)(9 22)(10 23)(11 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)
(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 40 11 30)(2 29 12 39)(3 38 13 28)(4 27 14 37)(5 36 15 26)(6 25 16 35)(7 34 17 24)(8 23 18 33)(9 32 19 22)(10 21 20 31)
G:=sub<Sym(40)| (1,19,17,15,13,11,9,7,5,3)(2,20,18,16,14,12,10,8,6,4)(21,23,25,27,29,31,33,35,37,39)(22,24,26,28,30,32,34,36,38,40), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33), (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), (1,40,11,30)(2,29,12,39)(3,38,13,28)(4,27,14,37)(5,36,15,26)(6,25,16,35)(7,34,17,24)(8,23,18,33)(9,32,19,22)(10,21,20,31)>;
G:=Group( (1,19,17,15,13,11,9,7,5,3)(2,20,18,16,14,12,10,8,6,4)(21,23,25,27,29,31,33,35,37,39)(22,24,26,28,30,32,34,36,38,40), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,21)(9,22)(10,23)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,31)(19,32)(20,33), (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), (1,40,11,30)(2,29,12,39)(3,38,13,28)(4,27,14,37)(5,36,15,26)(6,25,16,35)(7,34,17,24)(8,23,18,33)(9,32,19,22)(10,21,20,31) );
G=PermutationGroup([[(1,19,17,15,13,11,9,7,5,3),(2,20,18,16,14,12,10,8,6,4),(21,23,25,27,29,31,33,35,37,39),(22,24,26,28,30,32,34,36,38,40)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,21),(9,22),(10,23),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,31),(19,32),(20,33)], [(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)], [(1,40,11,30),(2,29,12,39),(3,38,13,28),(4,27,14,37),(5,36,15,26),(6,25,16,35),(7,34,17,24),(8,23,18,33),(9,32,19,22),(10,21,20,31)]])
58 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 | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20X |
| 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 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 10 | 10 | 50 | 1 | 1 | 10 | 10 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | D52 | C2×D52 | D10.9D10 |
| kernel | D10.9D10 | C52⋊2D4 | C5⋊D20 | C52⋊2Q8 | D5×C20 | C4×C5⋊D5 | C4×D5 | C52 | Dic5 | C20 | D10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 4 | 2 | 4 | 4 | 4 | 16 | 4 | 4 | 8 |
Matrix representation of D10.9D10 ►in GL4(𝔽41) generated by
| 34 | 7 | 0 | 0 |
| 34 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 14 | 11 | 0 | 0 |
| 27 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 40 | 6 |
| 0 | 0 | 35 | 35 |
| 30 | 9 | 0 | 0 |
| 32 | 11 | 0 | 0 |
| 0 | 0 | 40 | 6 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [34,34,0,0,7,1,0,0,0,0,1,0,0,0,0,1],[14,27,0,0,11,27,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,40,35,0,0,6,35],[30,32,0,0,9,11,0,0,0,0,40,0,0,0,6,1] >;
D10.9D10 in GAP, Magma, Sage, TeX
D_{10}._9D_{10} % in TeX
G:=Group("D10.9D10"); // GroupNames label
G:=SmallGroup(400,167);
// by ID
G=gap.SmallGroup(400,167);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=1,c^10=d^2=a^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^9>;
// generators/relations