direct product, metacyclic, supersoluble, monomial
Aliases: C5×Dic10, C52⋊3Q8, C20.7D5, C20.1C10, C10.17D10, Dic5.1C10, C5⋊(C5×Q8), C4.(C5×D5), (C5×C20).2C2, C2.3(D5×C10), C10.1(C2×C10), (C5×C10).6C22, (C5×Dic5).4C2, SmallGroup(200,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic10
G = < a,b,c | a5=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic10
►On 40 points
Generators in S40
(1 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)
(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 31 11 21)(2 30 12 40)(3 29 13 39)(4 28 14 38)(5 27 15 37)(6 26 16 36)(7 25 17 35)(8 24 18 34)(9 23 19 33)(10 22 20 32)
G:=sub<Sym(40)| (1,9,17,5,13)(2,10,18,6,14)(3,11,19,7,15)(4,12,20,8,16)(21,33,25,37,29)(22,34,26,38,30)(23,35,27,39,31)(24,36,28,40,32), (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), (1,31,11,21)(2,30,12,40)(3,29,13,39)(4,28,14,38)(5,27,15,37)(6,26,16,36)(7,25,17,35)(8,24,18,34)(9,23,19,33)(10,22,20,32)>;
G:=Group( (1,9,17,5,13)(2,10,18,6,14)(3,11,19,7,15)(4,12,20,8,16)(21,33,25,37,29)(22,34,26,38,30)(23,35,27,39,31)(24,36,28,40,32), (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), (1,31,11,21)(2,30,12,40)(3,29,13,39)(4,28,14,38)(5,27,15,37)(6,26,16,36)(7,25,17,35)(8,24,18,34)(9,23,19,33)(10,22,20,32) );
G=PermutationGroup([[(1,9,17,5,13),(2,10,18,6,14),(3,11,19,7,15),(4,12,20,8,16),(21,33,25,37,29),(22,34,26,38,30),(23,35,27,39,31),(24,36,28,40,32)], [(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)], [(1,31,11,21),(2,30,12,40),(3,29,13,39),(4,28,14,38),(5,27,15,37),(6,26,16,36),(7,25,17,35),(8,24,18,34),(9,23,19,33),(10,22,20,32)]])
C5×Dic10 is a maximal subgroup of
D20.D5 C52⋊4SD16 C52⋊2Q16 C52⋊3Q16 D20⋊D5 Dic10⋊D5 Dic10⋊5D5 C5×Q8×D5
65 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 20A | ··· | 20X | 20Y | ··· | 20AF |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | Q8 | D5 | D10 | Dic10 | C5×Q8 | C5×D5 | D5×C10 | C5×Dic10 |
| kernel | C5×Dic10 | C5×Dic5 | C5×C20 | Dic10 | Dic5 | C20 | C52 | C20 | C10 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C5×Dic10 ►in GL2(𝔽41) generated by
| 10 | 0 |
| 0 | 10 |
| 2 | 0 |
| 0 | 21 |
| 0 | 1 |
| 40 | 0 |
G:=sub<GL(2,GF(41))| [10,0,0,10],[2,0,0,21],[0,40,1,0] >;
C5×Dic10 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{10} % in TeX
G:=Group("C5xDic10"); // GroupNames label
G:=SmallGroup(200,27);
// by ID
G=gap.SmallGroup(200,27);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,100,221,106,4004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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