direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C10×Dic5, C10⋊2C20, C10.20D10, C102.1C2, (C5×C10)⋊5C4, C5⋊3(C2×C20), C22.(C5×D5), C52⋊11(C2×C4), C2.2(D5×C10), (C2×C10).5D5, (C2×C10).2C10, C10.4(C2×C10), (C5×C10).9C22, SmallGroup(200,30)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C10×Dic5 |
Generators and relations for C10×Dic5
G = < a,b,c | a10=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C10×Dic5
►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 21 9 29 7 27 5 25 3 23)(2 22 10 30 8 28 6 26 4 24)(11 38 13 40 15 32 17 34 19 36)(12 39 14 31 16 33 18 35 20 37)
(1 18 27 39)(2 19 28 40)(3 20 29 31)(4 11 30 32)(5 12 21 33)(6 13 22 34)(7 14 23 35)(8 15 24 36)(9 16 25 37)(10 17 26 38)
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), (1,21,9,29,7,27,5,25,3,23)(2,22,10,30,8,28,6,26,4,24)(11,38,13,40,15,32,17,34,19,36)(12,39,14,31,16,33,18,35,20,37), (1,18,27,39)(2,19,28,40)(3,20,29,31)(4,11,30,32)(5,12,21,33)(6,13,22,34)(7,14,23,35)(8,15,24,36)(9,16,25,37)(10,17,26,38)>;
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), (1,21,9,29,7,27,5,25,3,23)(2,22,10,30,8,28,6,26,4,24)(11,38,13,40,15,32,17,34,19,36)(12,39,14,31,16,33,18,35,20,37), (1,18,27,39)(2,19,28,40)(3,20,29,31)(4,11,30,32)(5,12,21,33)(6,13,22,34)(7,14,23,35)(8,15,24,36)(9,16,25,37)(10,17,26,38) );
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)], [(1,21,9,29,7,27,5,25,3,23),(2,22,10,30,8,28,6,26,4,24),(11,38,13,40,15,32,17,34,19,36),(12,39,14,31,16,33,18,35,20,37)], [(1,18,27,39),(2,19,28,40),(3,20,29,31),(4,11,30,32),(5,12,21,33),(6,13,22,34),(7,14,23,35),(8,15,24,36),(9,16,25,37),(10,17,26,38)]])
C10×Dic5 is a maximal subgroup of
D10⋊Dic5 C10.D20 Dic5⋊Dic5 C10.Dic10 C102.C4 Dic5.D10 D5×C2×C20
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 10A | ··· | 10L | 10M | ··· | 10AP | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 5 | ··· | 5 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C5 | C10 | C10 | C20 | D5 | Dic5 | D10 | C5×D5 | C5×Dic5 | D5×C10 |
| kernel | C10×Dic5 | C5×Dic5 | C102 | C5×C10 | C2×Dic5 | Dic5 | C2×C10 | C10 | C2×C10 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 4 | 8 | 4 | 16 | 2 | 4 | 2 | 8 | 16 | 8 |
Matrix representation of C10×Dic5 ►in GL3(𝔽41) generated by
| 31 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 25 | 0 |
| 0 | 0 | 23 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 40 | 0 |
G:=sub<GL(3,GF(41))| [31,0,0,0,18,0,0,0,18],[1,0,0,0,25,0,0,0,23],[1,0,0,0,0,40,0,1,0] >;
C10×Dic5 in GAP, Magma, Sage, TeX
C_{10}\times {\rm Dic}_5 % in TeX
G:=Group("C10xDic5"); // GroupNames label
G:=SmallGroup(200,30);
// by ID
G=gap.SmallGroup(200,30);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,100,4004]);
// Polycyclic
G:=Group<a,b,c|a^10=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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