metabelian, supersoluble, monomial, A-group
Aliases: C15⋊1F5, C52⋊5Dic3, (C5×C15)⋊3C4, C5⋊1(C3⋊F5), C3⋊(C5⋊F5), C5⋊D5.2S3, (C3×C5⋊D5).1C2, SmallGroup(300,34)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×C15 — C15⋊F5 |
Generators and relations for C15⋊F5
G = < a,b,c | a15=b5=c4=1, ab=ba, cac-1=a8, cbc-1=b3 >
Character table of C15⋊F5
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 6 | 15A | 15B | 15C | 15D | 15E | 15F | 15G | 15H | 15I | 15J | 15K | 15L | |
| size | 1 | 25 | 2 | 75 | 75 | 4 | 4 | 4 | 4 | 4 | 4 | 50 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | -2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ7 | 4 | 0 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | 4 | -1 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ8 | 4 | 0 | 4 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | orthogonal lifted from F5 |
| ρ9 | 4 | 0 | 4 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | 0 | 4 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 4 | -1 | 4 | -1 | -1 | orthogonal lifted from F5 |
| ρ11 | 4 | 0 | 4 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | -1 | 4 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ12 | 4 | 0 | 4 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 4 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ13 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | -2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | -2 | complex lifted from C3⋊F5 |
| ρ14 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | -2 | 1-√-15/2 | -2 | 1+√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ15 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | -2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | -2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ16 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | -2 | 1+√-15/2 | -2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ17 | 4 | 0 | -2 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 1+√-15/2 | -2 | 1-√-15/2 | 1-√-15/2 | -2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ18 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | 0 | -2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | -2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ19 | 4 | 0 | -2 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 1-√-15/2 | 1+√-15/2 | -2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | -2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ20 | 4 | 0 | -2 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 1-√-15/2 | -2 | 1+√-15/2 | 1+√-15/2 | -2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ21 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | 0 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | -2 | 1-√-15/2 | -2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ22 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | 0 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | -2 | 1+√-15/2 | -2 | 1-√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ23 | 4 | 0 | -2 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | 0 | 1+√-15/2 | 1-√-15/2 | -2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | -2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ24 | 4 | 0 | -2 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | 0 | 1-√-15/2 | 1+√-15/2 | 1-√-15/2 | -2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | -2 | complex lifted from C3⋊F5 |
►On 75 points
Generators in S75
(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)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)
(1 53 25 61 33)(2 54 26 62 34)(3 55 27 63 35)(4 56 28 64 36)(5 57 29 65 37)(6 58 30 66 38)(7 59 16 67 39)(8 60 17 68 40)(9 46 18 69 41)(10 47 19 70 42)(11 48 20 71 43)(12 49 21 72 44)(13 50 22 73 45)(14 51 23 74 31)(15 52 24 75 32)
(2 3 5 9)(4 7 13 10)(6 11)(8 15 14 12)(16 45 70 56)(17 32 74 49)(18 34 63 57)(19 36 67 50)(20 38 71 58)(21 40 75 51)(22 42 64 59)(23 44 68 52)(24 31 72 60)(25 33 61 53)(26 35 65 46)(27 37 69 54)(28 39 73 47)(29 41 62 55)(30 43 66 48)
G:=sub<Sym(75)| (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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75), (1,53,25,61,33)(2,54,26,62,34)(3,55,27,63,35)(4,56,28,64,36)(5,57,29,65,37)(6,58,30,66,38)(7,59,16,67,39)(8,60,17,68,40)(9,46,18,69,41)(10,47,19,70,42)(11,48,20,71,43)(12,49,21,72,44)(13,50,22,73,45)(14,51,23,74,31)(15,52,24,75,32), (2,3,5,9)(4,7,13,10)(6,11)(8,15,14,12)(16,45,70,56)(17,32,74,49)(18,34,63,57)(19,36,67,50)(20,38,71,58)(21,40,75,51)(22,42,64,59)(23,44,68,52)(24,31,72,60)(25,33,61,53)(26,35,65,46)(27,37,69,54)(28,39,73,47)(29,41,62,55)(30,43,66,48)>;
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)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75), (1,53,25,61,33)(2,54,26,62,34)(3,55,27,63,35)(4,56,28,64,36)(5,57,29,65,37)(6,58,30,66,38)(7,59,16,67,39)(8,60,17,68,40)(9,46,18,69,41)(10,47,19,70,42)(11,48,20,71,43)(12,49,21,72,44)(13,50,22,73,45)(14,51,23,74,31)(15,52,24,75,32), (2,3,5,9)(4,7,13,10)(6,11)(8,15,14,12)(16,45,70,56)(17,32,74,49)(18,34,63,57)(19,36,67,50)(20,38,71,58)(21,40,75,51)(22,42,64,59)(23,44,68,52)(24,31,72,60)(25,33,61,53)(26,35,65,46)(27,37,69,54)(28,39,73,47)(29,41,62,55)(30,43,66,48) );
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),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)], [(1,53,25,61,33),(2,54,26,62,34),(3,55,27,63,35),(4,56,28,64,36),(5,57,29,65,37),(6,58,30,66,38),(7,59,16,67,39),(8,60,17,68,40),(9,46,18,69,41),(10,47,19,70,42),(11,48,20,71,43),(12,49,21,72,44),(13,50,22,73,45),(14,51,23,74,31),(15,52,24,75,32)], [(2,3,5,9),(4,7,13,10),(6,11),(8,15,14,12),(16,45,70,56),(17,32,74,49),(18,34,63,57),(19,36,67,50),(20,38,71,58),(21,40,75,51),(22,42,64,59),(23,44,68,52),(24,31,72,60),(25,33,61,53),(26,35,65,46),(27,37,69,54),(28,39,73,47),(29,41,62,55),(30,43,66,48)]])
Matrix representation of C15⋊F5 ►in GL8(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 55 | 34 | 27 | 6 |
| 0 | 0 | 0 | 0 | 55 | 28 | 0 | 33 |
| 0 | 0 | 0 | 0 | 0 | 28 | 55 | 6 |
| 0 | 0 | 0 | 0 | 27 | 34 | 55 | 0 |
| 60 | 60 | 60 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,55,55,0,27,0,0,0,0,34,28,28,34,0,0,0,0,27,0,55,55,0,0,0,0,6,33,6,0],[60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
C15⋊F5 in GAP, Magma, Sage, TeX
C_{15}\rtimes F_5 % in TeX
G:=Group("C15:F5"); // GroupNames label
G:=SmallGroup(300,34);
// by ID
G=gap.SmallGroup(300,34);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,723,488,4504,3009]);
// Polycyclic
G:=Group<a,b,c|a^15=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^8,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C15⋊F5 in TeX
Character table of C15⋊F5 in TeX