direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×C5⋊C8, C5⋊C40, C10.C20, C52⋊2C8, C10.6F5, Dic5.2C10, C2.(C5×F5), (C5×C10).1C4, (C5×Dic5).1C2, SmallGroup(200,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5×C5⋊C8 |
Generators and relations for C5×C5⋊C8
G = < a,b,c | a5=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C5×C5⋊C8
►On 40 points
Generators in S40
(1 24 32 9 35)(2 17 25 10 36)(3 18 26 11 37)(4 19 27 12 38)(5 20 28 13 39)(6 21 29 14 40)(7 22 30 15 33)(8 23 31 16 34)
(1 24 32 9 35)(2 10 17 36 25)(3 37 11 26 18)(4 27 38 19 12)(5 20 28 13 39)(6 14 21 40 29)(7 33 15 30 22)(8 31 34 23 16)
(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)| (1,24,32,9,35)(2,17,25,10,36)(3,18,26,11,37)(4,19,27,12,38)(5,20,28,13,39)(6,21,29,14,40)(7,22,30,15,33)(8,23,31,16,34), (1,24,32,9,35)(2,10,17,36,25)(3,37,11,26,18)(4,27,38,19,12)(5,20,28,13,39)(6,14,21,40,29)(7,33,15,30,22)(8,31,34,23,16), (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( (1,24,32,9,35)(2,17,25,10,36)(3,18,26,11,37)(4,19,27,12,38)(5,20,28,13,39)(6,21,29,14,40)(7,22,30,15,33)(8,23,31,16,34), (1,24,32,9,35)(2,10,17,36,25)(3,37,11,26,18)(4,27,38,19,12)(5,20,28,13,39)(6,14,21,40,29)(7,33,15,30,22)(8,31,34,23,16), (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([[(1,24,32,9,35),(2,17,25,10,36),(3,18,26,11,37),(4,19,27,12,38),(5,20,28,13,39),(6,21,29,14,40),(7,22,30,15,33),(8,23,31,16,34)], [(1,24,32,9,35),(2,10,17,36,25),(3,37,11,26,18),(4,27,38,19,12),(5,20,28,13,39),(6,14,21,40,29),(7,33,15,30,22),(8,31,34,23,16)], [(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⋊C8 is a maximal subgroup of
Dic5.4F5 D10.F5 Dic5.F5
50 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10I | 20A | ··· | 20H | 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 | 40 | ··· | 40 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 | F5 | C5⋊C8 | C5×F5 | C5×C5⋊C8 |
| kernel | C5×C5⋊C8 | C5×Dic5 | C5×C10 | C5⋊C8 | C52 | Dic5 | C10 | C5 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C5⋊C8 ►in GL5(𝔽41)
| 18 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 0 |
| 0 | 0 | 37 | 0 | 0 |
| 0 | 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 0 | 37 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 | 37 |
| 0 | 0 | 10 | 0 | 28 |
| 0 | 0 | 0 | 16 | 20 |
| 0 | 0 | 0 | 0 | 18 |
| 38 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 1 | 21 |
| 0 | 3 | 0 | 0 | 16 |
| 0 | 32 | 1 | 0 | 34 |
| 0 | 36 | 0 | 0 | 14 |
G:=sub<GL(5,GF(41))| [18,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37],[1,0,0,0,0,0,37,0,0,0,0,0,10,0,0,0,0,0,16,0,0,37,28,20,18],[38,0,0,0,0,0,27,3,32,36,0,0,0,1,0,0,1,0,0,0,0,21,16,34,14] >;
C5×C5⋊C8 in GAP, Magma, Sage, TeX
C_5\times C_5\rtimes C_8
% in TeX
G:=Group("C5xC5:C8"); // GroupNames label
G:=SmallGroup(200,18);
// by ID
G=gap.SmallGroup(200,18);
# by ID
G:=PCGroup([5,-2,-5,-2,-2,-5,50,42,2004,414]);
// 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^3>;
// generators/relations
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