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## G = C15⋊F5order 300 = 22·3·52

### 1st semidirect product of C15 and F5 acting via F5/C5=C4

Aliases: C151F5, C525Dic3, (C5×C15)⋊3C4, C51(C3⋊F5), C3⋊(C5⋊F5), C5⋊D5.2S3, (C3×C5⋊D5).1C2, SmallGroup(300,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — C15⋊F5
 Chief series C1 — C5 — C52 — C5×C15 — C3×C5⋊D5 — C15⋊F5
 Lower central C5×C15 — C15⋊F5
 Upper central C1

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

Smallest permutation representation of C15⋊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 55 44 16 72)(2 56 45 17 73)(3 57 31 18 74)(4 58 32 19 75)(5 59 33 20 61)(6 60 34 21 62)(7 46 35 22 63)(8 47 36 23 64)(9 48 37 24 65)(10 49 38 25 66)(11 50 39 26 67)(12 51 40 27 68)(13 52 41 28 69)(14 53 42 29 70)(15 54 43 30 71)
(2 3 5 9)(4 7 13 10)(6 11)(8 15 14 12)(16 55 44 72)(17 57 33 65)(18 59 37 73)(19 46 41 66)(20 48 45 74)(21 50 34 67)(22 52 38 75)(23 54 42 68)(24 56 31 61)(25 58 35 69)(26 60 39 62)(27 47 43 70)(28 49 32 63)(29 51 36 71)(30 53 40 64)

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), (1,55,44,16,72)(2,56,45,17,73)(3,57,31,18,74)(4,58,32,19,75)(5,59,33,20,61)(6,60,34,21,62)(7,46,35,22,63)(8,47,36,23,64)(9,48,37,24,65)(10,49,38,25,66)(11,50,39,26,67)(12,51,40,27,68)(13,52,41,28,69)(14,53,42,29,70)(15,54,43,30,71), (2,3,5,9)(4,7,13,10)(6,11)(8,15,14,12)(16,55,44,72)(17,57,33,65)(18,59,37,73)(19,46,41,66)(20,48,45,74)(21,50,34,67)(22,52,38,75)(23,54,42,68)(24,56,31,61)(25,58,35,69)(26,60,39,62)(27,47,43,70)(28,49,32,63)(29,51,36,71)(30,53,40,64)>;

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), (1,55,44,16,72)(2,56,45,17,73)(3,57,31,18,74)(4,58,32,19,75)(5,59,33,20,61)(6,60,34,21,62)(7,46,35,22,63)(8,47,36,23,64)(9,48,37,24,65)(10,49,38,25,66)(11,50,39,26,67)(12,51,40,27,68)(13,52,41,28,69)(14,53,42,29,70)(15,54,43,30,71), (2,3,5,9)(4,7,13,10)(6,11)(8,15,14,12)(16,55,44,72)(17,57,33,65)(18,59,37,73)(19,46,41,66)(20,48,45,74)(21,50,34,67)(22,52,38,75)(23,54,42,68)(24,56,31,61)(25,58,35,69)(26,60,39,62)(27,47,43,70)(28,49,32,63)(29,51,36,71)(30,53,40,64) );

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)], [(1,55,44,16,72),(2,56,45,17,73),(3,57,31,18,74),(4,58,32,19,75),(5,59,33,20,61),(6,60,34,21,62),(7,46,35,22,63),(8,47,36,23,64),(9,48,37,24,65),(10,49,38,25,66),(11,50,39,26,67),(12,51,40,27,68),(13,52,41,28,69),(14,53,42,29,70),(15,54,43,30,71)], [(2,3,5,9),(4,7,13,10),(6,11),(8,15,14,12),(16,55,44,72),(17,57,33,65),(18,59,37,73),(19,46,41,66),(20,48,45,74),(21,50,34,67),(22,52,38,75),(23,54,42,68),(24,56,31,61),(25,58,35,69),(26,60,39,62),(27,47,43,70),(28,49,32,63),(29,51,36,71),(30,53,40,64)])

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

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