direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×Dic5, D10.2D5, C10.1D10, C2.1D52, C5⋊4(C4×D5), (C5×D5)⋊4C4, C52⋊7(C2×C4), C5⋊2(C2×Dic5), C52⋊6C4⋊1C2, (C5×Dic5)⋊2C2, (D5×C10).1C2, (C5×C10).1C22, SmallGroup(200,22)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — D5×Dic5 |
Generators and relations for D5×Dic5
G = < a,b,c,d | a5=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D5×Dic5
►On 40 points
Generators in S40
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)(21 27 23 29 25)(22 28 24 30 26)(31 35 39 33 37)(32 36 40 34 38)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)
(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 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)
G:=sub<Sym(40)| (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16)(21,27,23,29,25)(22,28,24,30,26)(31,35,39,33,37)(32,36,40,34,38), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(11,37)(12,38)(13,39)(14,40)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36), (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), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,40,30,35)>;
G:=Group( (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16)(21,27,23,29,25)(22,28,24,30,26)(31,35,39,33,37)(32,36,40,34,38), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,21)(8,22)(9,23)(10,24)(11,37)(12,38)(13,39)(14,40)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36), (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), (1,14,6,19)(2,13,7,18)(3,12,8,17)(4,11,9,16)(5,20,10,15)(21,34,26,39)(22,33,27,38)(23,32,28,37)(24,31,29,36)(25,40,30,35) );
G=PermutationGroup([[(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16),(21,27,23,29,25),(22,28,24,30,26),(31,35,39,33,37),(32,36,40,34,38)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,21),(8,22),(9,23),(10,24),(11,37),(12,38),(13,39),(14,40),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36)], [(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)], [(1,14,6,19),(2,13,7,18),(3,12,8,17),(4,11,9,16),(5,20,10,15),(21,34,26,39),(22,33,27,38),(23,32,28,37),(24,31,29,36),(25,40,30,35)]])
D5×Dic5 is a maximal subgroup of
D5.Dic10 D10.F5 D10.2F5 C52⋊4M4(2) D20⋊5D5 D20⋊D5 C4×D52 Dic5.D10 D10.4D10
D5×Dic5 is a maximal quotient of
C20.30D10 D10⋊Dic5 Dic5⋊Dic5
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 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 |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 25 | 25 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D5 | D5 | Dic5 | D10 | C4×D5 | D52 | D5×Dic5 |
| kernel | D5×Dic5 | C5×Dic5 | C52⋊6C4 | D5×C10 | C5×D5 | Dic5 | D10 | D5 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
Matrix representation of D5×Dic5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 40 |
| 0 | 0 | 35 | 35 |
| 35 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 13 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,6,35,0,0,40,35],[35,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[9,13,0,0,0,32,0,0,0,0,1,0,0,0,0,1] >;
D5×Dic5 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_5 % in TeX
G:=Group("D5xDic5"); // GroupNames label
G:=SmallGroup(200,22);
// by ID
G=gap.SmallGroup(200,22);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,26,328,4004]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^10=1,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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