direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D5×C10, C10⋊C10, C52⋊2C22, C5⋊(C2×C10), (C5×C10)⋊1C2, SmallGroup(100,14)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C10 |
Generators and relations for D5×C10
G = < a,b,c | a10=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Permutation representations of D5×C10
►On 20 points - transitive group 20T24
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 11)(8 12)(9 13)(10 14)
G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,11)(8,12)(9,13)(10,14)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,11)(8,12)(9,13)(10,14) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,11),(8,12),(9,13),(10,14)]])
G:=TransitiveGroup(20,24);
D5×C10 is a maximal subgroup of
C52⋊2D4 C5⋊D20
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | ··· | 10V |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D5 | D10 | C5×D5 | D5×C10 |
| kernel | D5×C10 | C5×D5 | C5×C10 | D10 | D5 | C10 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 2 | 2 | 8 | 8 |
Matrix representation of D5×C10 ►in GL2(𝔽11) generated by
| 2 | 0 |
| 0 | 2 |
| 3 | 0 |
| 5 | 4 |
| 7 | 8 |
| 5 | 4 |
G:=sub<GL(2,GF(11))| [2,0,0,2],[3,5,0,4],[7,5,8,4] >;
D5×C10 in GAP, Magma, Sage, TeX
D_5\times C_{10} % in TeX
G:=Group("D5xC10"); // GroupNames label
G:=SmallGroup(100,14);
// by ID
G=gap.SmallGroup(100,14);
# by ID
G:=PCGroup([4,-2,-2,-5,-5,1283]);
// Polycyclic
G:=Group<a,b,c|a^10=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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