direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C20, C20⋊2C10, Dic5⋊2C10, D10.2C10, C10.18D10, C5⋊2(C2×C20), (C5×C20)⋊3C2, C52⋊9(C2×C4), C2.1(D5×C10), C10.2(C2×C10), (C5×Dic5)⋊5C2, (D5×C10).4C2, (C5×C10).7C22, SmallGroup(200,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C20 |
Generators and relations for D5×C20
G = < a,b,c | a20=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C20
►On 40 points
Generators in S40
(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 17 13 9 5)(2 18 14 10 6)(3 19 15 11 7)(4 20 16 12 8)(21 25 29 33 37)(22 26 30 34 38)(23 27 31 35 39)(24 28 32 36 40)
(1 37)(2 38)(3 39)(4 40)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)
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), (1,17,13,9,5)(2,18,14,10,6)(3,19,15,11,7)(4,20,16,12,8)(21,25,29,33,37)(22,26,30,34,38)(23,27,31,35,39)(24,28,32,36,40), (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)>;
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), (1,17,13,9,5)(2,18,14,10,6)(3,19,15,11,7)(4,20,16,12,8)(21,25,29,33,37)(22,26,30,34,38)(23,27,31,35,39)(24,28,32,36,40), (1,37)(2,38)(3,39)(4,40)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36) );
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)], [(1,17,13,9,5),(2,18,14,10,6),(3,19,15,11,7),(4,20,16,12,8),(21,25,29,33,37),(22,26,30,34,38),(23,27,31,35,39),(24,28,32,36,40)], [(1,37),(2,38),(3,39),(4,40),(5,21),(6,22),(7,23),(8,24),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36)]])
D5×C20 is a maximal subgroup of
C20.30D10 C20.14F5 C20.12F5 C20⋊5F5 D20⋊5D5 D10.9D10 Dic10⋊5D5
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | ··· | 20AJ |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | D5 | D10 | C4×D5 | C5×D5 | D5×C10 | D5×C20 |
| kernel | D5×C20 | C5×Dic5 | C5×C20 | D5×C10 | C5×D5 | C4×D5 | Dic5 | C20 | D10 | D5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of D5×C20 ►in GL2(𝔽41) generated by
| 2 | 0 |
| 0 | 2 |
| 18 | 0 |
| 0 | 16 |
| 0 | 16 |
| 18 | 0 |
G:=sub<GL(2,GF(41))| [2,0,0,2],[18,0,0,16],[0,18,16,0] >;
D5×C20 in GAP, Magma, Sage, TeX
D_5\times C_{20} % in TeX
G:=Group("D5xC20"); // GroupNames label
G:=SmallGroup(200,28);
// by ID
G=gap.SmallGroup(200,28);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,106,4004]);
// Polycyclic
G:=Group<a,b,c|a^20=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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