direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C15×F5, C5⋊C60, D5.C30, C15⋊2C20, C52⋊3C12, (C5×C15)⋊5C4, (C5×D5).1C6, (C3×D5).2C10, (D5×C15).1C2, SmallGroup(300,28)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C15×F5 |
Generators and relations for C15×F5
G = < a,b,c | a15=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C15×F5
►On 60 points
Generators in S60
(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 41 42 43 44 45)(46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)
(1 4 7 10 13)(2 5 8 11 14)(3 6 9 12 15)(16 25 19 28 22)(17 26 20 29 23)(18 27 21 30 24)(31 37 43 34 40)(32 38 44 35 41)(33 39 45 36 42)(46 58 55 52 49)(47 59 56 53 50)(48 60 57 54 51)
(1 29 59 35)(2 30 60 36)(3 16 46 37)(4 17 47 38)(5 18 48 39)(6 19 49 40)(7 20 50 41)(8 21 51 42)(9 22 52 43)(10 23 53 44)(11 24 54 45)(12 25 55 31)(13 26 56 32)(14 27 57 33)(15 28 58 34)
G:=sub<Sym(60)| (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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)(31,37,43,34,40)(32,38,44,35,41)(33,39,45,36,42)(46,58,55,52,49)(47,59,56,53,50)(48,60,57,54,51), (1,29,59,35)(2,30,60,36)(3,16,46,37)(4,17,47,38)(5,18,48,39)(6,19,49,40)(7,20,50,41)(8,21,51,42)(9,22,52,43)(10,23,53,44)(11,24,54,45)(12,25,55,31)(13,26,56,32)(14,27,57,33)(15,28,58,34)>;
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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45)(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)(31,37,43,34,40)(32,38,44,35,41)(33,39,45,36,42)(46,58,55,52,49)(47,59,56,53,50)(48,60,57,54,51), (1,29,59,35)(2,30,60,36)(3,16,46,37)(4,17,47,38)(5,18,48,39)(6,19,49,40)(7,20,50,41)(8,21,51,42)(9,22,52,43)(10,23,53,44)(11,24,54,45)(12,25,55,31)(13,26,56,32)(14,27,57,33)(15,28,58,34) );
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),(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45),(46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)], [(1,4,7,10,13),(2,5,8,11,14),(3,6,9,12,15),(16,25,19,28,22),(17,26,20,29,23),(18,27,21,30,24),(31,37,43,34,40),(32,38,44,35,41),(33,39,45,36,42),(46,58,55,52,49),(47,59,56,53,50),(48,60,57,54,51)], [(1,29,59,35),(2,30,60,36),(3,16,46,37),(4,17,47,38),(5,18,48,39),(6,19,49,40),(7,20,50,41),(8,21,51,42),(9,22,52,43),(10,23,53,44),(11,24,54,45),(12,25,55,31),(13,26,56,32),(14,27,57,33),(15,28,58,34)]])
75 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 6A | 6B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 15I | ··· | 15R | 20A | ··· | 20H | 30A | ··· | 30H | 60A | ··· | 60P |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 6 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 5 | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 1 | ··· | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 | 5 | ··· | 5 |
75 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C3 | C4 | C5 | C6 | C10 | C12 | C15 | C20 | C30 | C60 | F5 | C3×F5 | C5×F5 | C15×F5 |
| kernel | C15×F5 | D5×C15 | C5×F5 | C5×C15 | C3×F5 | C5×D5 | C3×D5 | C52 | F5 | C15 | D5 | C5 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C15×F5 ►in GL4(𝔽61) generated by
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 0 | 22 |
| 34 | 0 | 0 | 31 |
| 0 | 9 | 0 | 41 |
| 0 | 0 | 58 | 3 |
| 0 | 0 | 0 | 20 |
| 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 60 | 1 | 0 | 0 |
| 33 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [22,0,0,0,0,22,0,0,0,0,22,0,0,0,0,22],[34,0,0,0,0,9,0,0,0,0,58,0,31,41,3,20],[1,1,60,33,0,0,1,0,1,0,0,0,0,0,0,60] >;
C15×F5 in GAP, Magma, Sage, TeX
C_{15}\times F_5 % in TeX
G:=Group("C15xF5"); // GroupNames label
G:=SmallGroup(300,28);
// by ID
G=gap.SmallGroup(300,28);
# by ID
G:=PCGroup([5,-2,-3,-5,-2,-5,150,3004,219]);
// Polycyclic
G:=Group<a,b,c|a^15=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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