direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C10×F5, C10⋊C20, D5⋊C20, D10.C10, C5⋊(C2×C20), (C5×C10)⋊1C4, (C5×D5)⋊3C4, D5.(C2×C10), C52⋊3(C2×C4), (D5×C10).2C2, (C5×D5).2C22, SmallGroup(200,45)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C10×F5 |
Generators and relations for C10×F5
G = < a,b,c | a10=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C10×F5
►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 3 5 7 9)(2 4 6 8 10)(11 17 13 19 15)(12 18 14 20 16)(21 29 27 25 23)(22 30 28 26 24)(31 35 39 33 37)(32 36 40 34 38)
(1 11 25 32)(2 12 26 33)(3 13 27 34)(4 14 28 35)(5 15 29 36)(6 16 30 37)(7 17 21 38)(8 18 22 39)(9 19 23 40)(10 20 24 31)
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), (1,3,5,7,9)(2,4,6,8,10)(11,17,13,19,15)(12,18,14,20,16)(21,29,27,25,23)(22,30,28,26,24)(31,35,39,33,37)(32,36,40,34,38), (1,11,25,32)(2,12,26,33)(3,13,27,34)(4,14,28,35)(5,15,29,36)(6,16,30,37)(7,17,21,38)(8,18,22,39)(9,19,23,40)(10,20,24,31)>;
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), (1,3,5,7,9)(2,4,6,8,10)(11,17,13,19,15)(12,18,14,20,16)(21,29,27,25,23)(22,30,28,26,24)(31,35,39,33,37)(32,36,40,34,38), (1,11,25,32)(2,12,26,33)(3,13,27,34)(4,14,28,35)(5,15,29,36)(6,16,30,37)(7,17,21,38)(8,18,22,39)(9,19,23,40)(10,20,24,31) );
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)], [(1,3,5,7,9),(2,4,6,8,10),(11,17,13,19,15),(12,18,14,20,16),(21,29,27,25,23),(22,30,28,26,24),(31,35,39,33,37),(32,36,40,34,38)], [(1,11,25,32),(2,12,26,33),(3,13,27,34),(4,14,28,35),(5,15,29,36),(6,16,30,37),(7,17,21,38),(8,18,22,39),(9,19,23,40),(10,20,24,31)]])
C10×F5 is a maximal subgroup of
D5.D20 D5.Dic10
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 10A | 10B | 10C | 10D | 10E | ··· | 10I | 10J | ··· | 10Q | 20A | ··· | 20P |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 5 | 5 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | F5 | C2×F5 | C5×F5 | C10×F5 |
| kernel | C10×F5 | C5×F5 | D5×C10 | C5×D5 | C5×C10 | C2×F5 | F5 | D10 | D5 | C10 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 4 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C10×F5 ►in GL5(𝔽41)
| 25 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 | 35 |
| 0 | 0 | 37 | 0 | 20 |
| 0 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 16 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 1 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 9 | 0 | 0 | 9 |
G:=sub<GL(5,GF(41))| [25,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,10,0,0,0,0,0,37,0,0,0,0,0,18,0,0,35,20,18,16],[40,0,0,0,0,0,32,9,1,9,0,0,0,1,0,0,1,0,0,0,0,0,0,0,9] >;
C10×F5 in GAP, Magma, Sage, TeX
C_{10}\times F_5 % in TeX
G:=Group("C10xF5"); // GroupNames label
G:=SmallGroup(200,45);
// by ID
G=gap.SmallGroup(200,45);
# by ID
G:=PCGroup([5,-2,-2,-5,-2,-5,100,2004,219]);
// Polycyclic
G:=Group<a,b,c|a^10=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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