metabelian, supersoluble, monomial
Aliases: Dic10⋊5D5, C20.27D10, D10.10D10, Dic5.17D10, C4.7D52, (C4×D5)⋊2D5, (D5×C20)⋊3C2, C5⋊1(C4○D20), C20⋊D5⋊4C2, C5⋊D20⋊4C2, C52⋊5(C4○D4), C5⋊1(Q8⋊2D5), (C5×Dic10)⋊7C2, (C5×C10).6C23, C10.6(C22×D5), Dic5⋊2D5⋊1C2, (C5×C20).20C22, (D5×C10).11C22, (C5×Dic5).4C22, C2.9(C2×D52), (C2×C5⋊D5).1C22, SmallGroup(400,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊5D5
G = < a,b,c,d | a20=c5=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >
Subgroups: 692 in 96 conjugacy classes, 32 normal (22 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×Q8, C5×D5, C5⋊D5, C5×C10, C4○D20, Q8⋊2D5, C5×Dic5, C5×Dic5, C5×C20, D5×C10, C2×C5⋊D5, Dic5⋊2D5, C5⋊D20, C5×Dic10, D5×C20, C20⋊D5, Dic10⋊5D5
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C4○D20, Q8⋊2D5, D52, C2×D52, Dic10⋊5D5
►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 11 35)(2 24 12 34)(3 23 13 33)(4 22 14 32)(5 21 15 31)(6 40 16 30)(7 39 17 29)(8 38 18 28)(9 37 19 27)(10 36 20 26)
(1 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 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), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26), (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28), (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)>;
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), (1,25,11,35)(2,24,12,34)(3,23,13,33)(4,22,14,32)(5,21,15,31)(6,40,16,30)(7,39,17,29)(8,38,18,28)(9,37,19,27)(10,36,20,26), (1,5,9,13,17)(2,6,10,14,18)(3,7,11,15,19)(4,8,12,16,20)(21,37,33,29,25)(22,38,34,30,26)(23,39,35,31,27)(24,40,36,32,28), (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38) );
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)], [(1,25,11,35),(2,24,12,34),(3,23,13,33),(4,22,14,32),(5,21,15,31),(6,40,16,30),(7,39,17,29),(8,38,18,28),(9,37,19,27),(10,36,20,26)], [(1,5,9,13,17),(2,6,10,14,18),(3,7,11,15,19),(4,8,12,16,20),(21,37,33,29,25),(22,38,34,30,26),(23,39,35,31,27),(24,40,36,32,28)], [(1,39),(2,40),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38)]])
52 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 | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 20Q | 20R | 20S | 20T | 20U | 20V |
| 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 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 10 | 50 | 50 | 2 | 5 | 5 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | Q8⋊2D5 | D52 | C2×D52 | Dic10⋊5D5 |
| kernel | Dic10⋊5D5 | Dic5⋊2D5 | C5⋊D20 | C5×Dic10 | D5×C20 | C20⋊D5 | Dic10 | C4×D5 | C52 | Dic5 | C20 | D10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 4 | 2 | 8 | 2 | 4 | 4 | 8 |
Matrix representation of Dic10⋊5D5 ►in GL4(𝔽41) generated by
| 14 | 30 | 0 | 0 |
| 11 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 40 | 0 | 0 |
| 3 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 6 |
| 17 | 40 | 0 | 0 |
| 1 | 24 | 0 | 0 |
| 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [14,11,0,0,30,9,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,6],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,6,40] >;
Dic10⋊5D5 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes_5D_5 % in TeX
G:=Group("Dic10:5D5"); // GroupNames label
G:=SmallGroup(400,168);
// by ID
G=gap.SmallGroup(400,168);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,55,116,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^20=c^5=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations