metabelian, supersoluble, monomial
Aliases: Dic10⋊4D5, C20.23D10, Dic5.2D10, C4.12D52, C5⋊D5⋊2Q8, C5⋊1(Q8×D5), C52⋊3(C2×Q8), C52⋊2Q8⋊3C2, (C5×Dic10)⋊6C2, (C5×C10).4C23, C10.4(C22×D5), (C5×C20).19C22, Dic5⋊2D5.1C2, (C5×Dic5).3C22, C52⋊6C4.10C22, C2.7(C2×D52), (C4×C5⋊D5).1C2, (C2×C5⋊D5).13C22, SmallGroup(400,166)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic10⋊D5
G = < a,b,c,d | a20=c5=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a9, bc=cb, bd=db, dcd=c-1 >
Subgroups: 556 in 92 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, Q8, D5, C10, C10, C2×Q8, Dic5, Dic5, C20, C20, D10, C52, Dic10, Dic10, C4×D5, C5×Q8, C5⋊D5, C5×C10, Q8×D5, C5×Dic5, C52⋊6C4, C5×C20, C2×C5⋊D5, Dic5⋊2D5, C52⋊2Q8, C5×Dic10, C4×C5⋊D5, Dic10⋊D5
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, C22×D5, Q8×D5, D52, C2×D52, Dic10⋊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 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 17)(2 6)(3 15)(5 13)(7 11)(8 20)(10 18)(12 16)(21 33)(23 31)(24 40)(25 29)(26 38)(28 36)(30 34)(35 39)
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), (1,17)(2,6)(3,15)(5,13)(7,11)(8,20)(10,18)(12,16)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39)>;
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), (1,17)(2,6)(3,15)(5,13)(7,11)(8,20)(10,18)(12,16)(21,33)(23,31)(24,40)(25,29)(26,38)(28,36)(30,34)(35,39) );
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)], [(1,17),(2,6),(3,15),(5,13),(7,11),(8,20),(10,18),(12,16),(21,33),(23,31),(24,40),(25,29),(26,38),(28,36),(30,34),(35,39)]])
46 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | ··· | 20L | 20M | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 25 | 25 | 2 | 10 | 10 | 10 | 10 | 50 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 20 | ··· | 20 |
46 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | Q8×D5 | D52 | C2×D52 | Dic10⋊D5 |
| kernel | Dic10⋊D5 | Dic5⋊2D5 | C52⋊2Q8 | C5×Dic10 | C4×C5⋊D5 | C5⋊D5 | Dic10 | Dic5 | C20 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 8 | 4 | 4 | 4 | 4 | 8 |
Matrix representation of Dic10⋊D5 ►in GL6(𝔽41)
| 34 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 34 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [34,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6],[1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,6,40] >;
Dic10⋊D5 in GAP, Magma, Sage, TeX
{\rm Dic}_{10}\rtimes D_5 % in TeX
G:=Group("Dic10:D5"); // GroupNames label
G:=SmallGroup(400,166);
// by ID
G=gap.SmallGroup(400,166);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,55,218,116,50,970,11525]);
// 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,d*a*d=a^9,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations