direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×C5⋊2C8, C5⋊2C40, C52⋊6C8, C20.8D5, C20.2C10, C10.2C20, C10.5Dic5, C4.2(C5×D5), C2.(C5×Dic5), (C5×C10).5C4, (C5×C20).3C2, SmallGroup(200,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5×C5⋊2C8 |
Generators and relations for C5×C5⋊2C8
G = < a,b,c | a5=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C5⋊2C8
►On 40 points
Generators in S40
(1 26 22 9 35)(2 27 23 10 36)(3 28 24 11 37)(4 29 17 12 38)(5 30 18 13 39)(6 31 19 14 40)(7 32 20 15 33)(8 25 21 16 34)
(1 26 22 9 35)(2 36 10 23 27)(3 28 24 11 37)(4 38 12 17 29)(5 30 18 13 39)(6 40 14 19 31)(7 32 20 15 33)(8 34 16 21 25)
(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)
G:=sub<Sym(40)| (1,26,22,9,35)(2,27,23,10,36)(3,28,24,11,37)(4,29,17,12,38)(5,30,18,13,39)(6,31,19,14,40)(7,32,20,15,33)(8,25,21,16,34), (1,26,22,9,35)(2,36,10,23,27)(3,28,24,11,37)(4,38,12,17,29)(5,30,18,13,39)(6,40,14,19,31)(7,32,20,15,33)(8,34,16,21,25), (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)>;
G:=Group( (1,26,22,9,35)(2,27,23,10,36)(3,28,24,11,37)(4,29,17,12,38)(5,30,18,13,39)(6,31,19,14,40)(7,32,20,15,33)(8,25,21,16,34), (1,26,22,9,35)(2,36,10,23,27)(3,28,24,11,37)(4,38,12,17,29)(5,30,18,13,39)(6,40,14,19,31)(7,32,20,15,33)(8,34,16,21,25), (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) );
G=PermutationGroup([[(1,26,22,9,35),(2,27,23,10,36),(3,28,24,11,37),(4,29,17,12,38),(5,30,18,13,39),(6,31,19,14,40),(7,32,20,15,33),(8,25,21,16,34)], [(1,26,22,9,35),(2,36,10,23,27),(3,28,24,11,37),(4,38,12,17,29),(5,30,18,13,39),(6,40,14,19,31),(7,32,20,15,33),(8,34,16,21,25)], [(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)]])
C5×C5⋊2C8 is a maximal subgroup of
C52⋊3C16 C20.29D10 C20.30D10 C20.31D10 C5⋊D40 C52⋊3SD16 C52⋊4SD16 C52⋊3Q16 D5×C40
80 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20AB | 40A | ··· | 40P |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 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 | C4 | C5 | C8 | C10 | C20 | C40 | D5 | Dic5 | C5⋊2C8 | C5×D5 | C5×Dic5 | C5×C5⋊2C8 |
| kernel | C5×C5⋊2C8 | C5×C20 | C5×C10 | C5⋊2C8 | C52 | C20 | C10 | C5 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of C5×C5⋊2C8 ►in GL2(𝔽41) generated by
| 16 | 0 |
| 0 | 16 |
| 16 | 0 |
| 0 | 18 |
| 0 | 1 |
| 32 | 0 |
G:=sub<GL(2,GF(41))| [16,0,0,16],[16,0,0,18],[0,32,1,0] >;
C5×C5⋊2C8 in GAP, Magma, Sage, TeX
C_5\times C_5\rtimes_2C_8
% in TeX
G:=Group("C5xC5:2C8"); // GroupNames label
G:=SmallGroup(200,15);
// by ID
G=gap.SmallGroup(200,15);
# by ID
G:=PCGroup([5,-2,-5,-2,-2,-5,50,42,4004]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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