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G = C15⋊F5order 300 = 22·3·52

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

metabelian, supersoluble, monomial, A-group

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

C1C5×C15 — C15⋊F5
C1C5C52C5×C15C3×C5⋊D5 — C15⋊F5
C5×C15 — C15⋊F5
C1

Generators and relations for C15⋊F5
 G = < a,b,c | a15=b5=c4=1, ab=ba, cac-1=a8, cbc-1=b3 >

25C2
75C4
25C6
5D5
5D5
5D5
5D5
5D5
5D5
25Dic3
15F5
15F5
15F5
15F5
15F5
15F5
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C3×D5
5C3⋊F5
5C3⋊F5
5C3⋊F5
5C3⋊F5
5C3⋊F5
5C3⋊F5
3C5⋊F5

Character table of C15⋊F5

 class 1234A4B5A5B5C5D5E5F615A15B15C15D15E15F15G15H15I15J15K15L
 size 1252757544444450444444444444
ρ1111111111111111111111111    trivial
ρ2111-1-11111111111111111111    linear of order 2
ρ31-11-ii111111-1111111111111    linear of order 4
ρ41-11i-i111111-1111111111111    linear of order 4
ρ522-100222222-1-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-2-1002222221-1-1-1-1-1-1-1-1-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ740400-1-1-1-14-104-1-1-1-14-1-1-1-1-1-1    orthogonal lifted from F5
ρ840400-1-1-14-1-10-1-1-14-1-1-1-1-1-1-14    orthogonal lifted from F5
ρ940400-1-14-1-1-10-1-1-1-1-1-1-1-14-14-1    orthogonal lifted from F5
ρ1040400-1-1-1-1-140-1-1-1-1-1-1-14-14-1-1    orthogonal lifted from F5
ρ11404004-1-1-1-1-10-14-1-14-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1240400-14-1-1-1-10-1-14-1-1-14-1-1-1-1-1    orthogonal lifted from F5
ρ1340-200-1-1-14-1-101+-15/21--15/21+-15/2-21+-15/21--15/21--15/21+-15/21+-15/21--15/21--15/2-2    complex lifted from C3⋊F5
ρ1440-200-1-1-1-1-1401--15/21--15/21+-15/21--15/21+-15/21+-15/21--15/2-21--15/2-21+-15/21+-15/2    complex lifted from C3⋊F5
ρ1540-200-1-1-1-14-10-21--15/21--15/21--15/21+-15/2-21+-15/21--15/21+-15/21+-15/21--15/21+-15/2    complex lifted from C3⋊F5
ρ1640-200-1-14-1-1-101+-15/21+-15/21+-15/21--15/21--15/21--15/21--15/21--15/2-21+-15/2-21+-15/2    complex lifted from C3⋊F5
ρ1740-2004-1-1-1-1-101+-15/2-21--15/21--15/2-21--15/21+-15/21+-15/21--15/21--15/21+-15/21+-15/2    complex lifted from C3⋊F5
ρ1840-200-1-1-1-14-10-21+-15/21+-15/21+-15/21--15/2-21--15/21+-15/21--15/21--15/21+-15/21--15/2    complex lifted from C3⋊F5
ρ1940-200-14-1-1-1-101--15/21+-15/2-21--15/21--15/21+-15/2-21+-15/21+-15/21--15/21--15/21+-15/2    complex lifted from C3⋊F5
ρ2040-2004-1-1-1-1-101--15/2-21+-15/21+-15/2-21+-15/21--15/21--15/21+-15/21+-15/21--15/21--15/2    complex lifted from C3⋊F5
ρ2140-200-1-14-1-1-101--15/21--15/21--15/21+-15/21+-15/21+-15/21+-15/21+-15/2-21--15/2-21--15/2    complex lifted from C3⋊F5
ρ2240-200-1-1-1-1-1401+-15/21+-15/21--15/21+-15/21--15/21--15/21+-15/2-21+-15/2-21--15/21--15/2    complex lifted from C3⋊F5
ρ2340-200-14-1-1-1-101+-15/21--15/2-21+-15/21+-15/21--15/2-21--15/21--15/21+-15/21+-15/21--15/2    complex lifted from C3⋊F5
ρ2440-200-1-1-14-1-101--15/21+-15/21--15/2-21--15/21+-15/21+-15/21--15/21--15/21+-15/21+-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)

10000000
01000000
00100000
00010000
00005534276
00005528033
0000028556
00002734550
,
606060600000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
10000000
00010000
01000000
606060600000
00000010
00001000
00000001
00000100

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

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