direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×C3⋊F5, C15⋊7F5, C15⋊1C20, C52⋊3Dic3, C3⋊(C5×F5), (C5×C15)⋊2C4, D5.(C5×S3), C5⋊(C5×Dic3), (C5×D5).1S3, (D5×C15).4C2, (C3×D5).1C10, SmallGroup(300,32)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C5×C3⋊F5 |
Generators and relations for C5×C3⋊F5
G = < a,b,c,d | a5=b3=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C5×C3⋊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 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)
(1 5 4 3 2)(6 10 9 8 7)(11 15 14 13 12)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)(31 33 35 32 34)(36 38 40 37 39)(41 43 45 42 44)(46 49 47 50 48)(51 54 52 55 53)(56 59 57 60 58)
(1 31 16 46)(2 32 17 47)(3 33 18 48)(4 34 19 49)(5 35 20 50)(6 36 21 51)(7 37 22 52)(8 38 23 53)(9 39 24 54)(10 40 25 55)(11 41 26 56)(12 42 27 57)(13 43 28 58)(14 44 29 59)(15 45 30 60)
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), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,33,35,32,34)(36,38,40,37,39)(41,43,45,42,44)(46,49,47,50,48)(51,54,52,55,53)(56,59,57,60,58), (1,31,16,46)(2,32,17,47)(3,33,18,48)(4,34,19,49)(5,35,20,50)(6,36,21,51)(7,37,22,52)(8,38,23,53)(9,39,24,54)(10,40,25,55)(11,41,26,56)(12,42,27,57)(13,43,28,58)(14,44,29,59)(15,45,30,60)>;
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), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30)(31,41,36)(32,42,37)(33,43,38)(34,44,39)(35,45,40)(46,56,51)(47,57,52)(48,58,53)(49,59,54)(50,60,55), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30)(31,33,35,32,34)(36,38,40,37,39)(41,43,45,42,44)(46,49,47,50,48)(51,54,52,55,53)(56,59,57,60,58), (1,31,16,46)(2,32,17,47)(3,33,18,48)(4,34,19,49)(5,35,20,50)(6,36,21,51)(7,37,22,52)(8,38,23,53)(9,39,24,54)(10,40,25,55)(11,41,26,56)(12,42,27,57)(13,43,28,58)(14,44,29,59)(15,45,30,60) );
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)], [(1,6,11),(2,7,12),(3,8,13),(4,9,14),(5,10,15),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30),(31,41,36),(32,42,37),(33,43,38),(34,44,39),(35,45,40),(46,56,51),(47,57,52),(48,58,53),(49,59,54),(50,60,55)], [(1,5,4,3,2),(6,10,9,8,7),(11,15,14,13,12),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30),(31,33,35,32,34),(36,38,40,37,39),(41,43,45,42,44),(46,49,47,50,48),(51,54,52,55,53),(56,59,57,60,58)], [(1,31,16,46),(2,32,17,47),(3,33,18,48),(4,34,19,49),(5,35,20,50),(6,36,21,51),(7,37,22,52),(8,38,23,53),(9,39,24,54),(10,40,25,55),(11,41,26,56),(12,42,27,57),(13,43,28,58),(14,44,29,59),(15,45,30,60)]])
45 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 6 | 10A | 10B | 10C | 10D | 15A | 15B | 15C | 15D | 15E | ··· | 15N | 20A | ··· | 20H | 30A | 30B | 30C | 30D |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 6 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | 30 | 30 | 30 |
| size | 1 | 5 | 2 | 15 | 15 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 10 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 15 | ··· | 15 | 10 | 10 | 10 | 10 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | C5×S3 | C5×Dic3 | F5 | C3⋊F5 | C5×F5 | C5×C3⋊F5 |
| kernel | C5×C3⋊F5 | D5×C15 | C5×C15 | C3⋊F5 | C3×D5 | C15 | C5×D5 | C52 | D5 | C5 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 4 | 4 | 1 | 2 | 4 | 8 |
Matrix representation of C5×C3⋊F5 ►in GL4(𝔽61) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 13 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 |
| 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 47 |
| 58 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 34 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[13,0,0,0,0,13,0,0,0,0,47,0,0,0,0,47],[58,0,0,0,0,20,0,0,0,0,34,0,0,0,0,9],[0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0] >;
C5×C3⋊F5 in GAP, Magma, Sage, TeX
C_5\times C_3\rtimes F_5
% in TeX
G:=Group("C5xC3:F5"); // GroupNames label
G:=SmallGroup(300,32);
// by ID
G=gap.SmallGroup(300,32);
# by ID
G:=PCGroup([5,-2,-5,-2,-3,-5,50,803,4504,614]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^3=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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