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G = C5×F9  order 360 = 23·32·5

Direct product of C5 and F9

direct product, metabelian, soluble, monomial, A-group

Aliases: C5×F9, C32⋊C40, C3⋊S3.C20, (C3×C15)⋊1C8, C32⋊C4.1C10, (C5×C3⋊S3).1C4, (C5×C32⋊C4).1C2, SmallGroup(360,123)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C5×F9
Chief series
C1 — C32 — C3⋊S3 — C32⋊C4 — C5×C32⋊C4 — C5×F9
Lower central
C32 — C5×F9
Upper central
C1 — C5

Generators and relations for C5×F9
 G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

C1
9C2
4C3
C5
9C4
12S3
C32
9C10
4C15
9C8
C3⋊S3
9C20
12C5×S3
C3×C15
C32⋊C4
9C40
C5×C3⋊S3
F9
C5×C32⋊C4
C5×F9

Smallest permutation representation of C5×F9
►On 45 points
Generators in S45
(1 2 4 3 5)(6 21 43 26 37)(7 14 44 27 30)(8 15 45 28 31)(9 16 38 29 32)(10 17 39 22 33)(11 18 40 23 34)(12 19 41 24 35)(13 20 42 25 36)

(1 19 15)(2 41 45)(3 35 31)(4 24 28)(5 12 8)(6 7 9)(10 13 11)(14 16 21)(17 20 18)(22 25 23)(26 27 29)(30 32 37)(33 36 34)(38 43 44)(39 42 40)

(1 20 16)(2 42 38)(3 36 32)(4 25 29)(5 13 9)(6 12 11)(7 8 10)(14 15 17)(18 21 19)(22 27 28)(23 26 24)(30 31 33)(34 37 35)(39 44 45)(40 43 41)

(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)
 
G:=sub<Sym(45)| (1,2,4,3,5)(6,21,43,26,37)(7,14,44,27,30)(8,15,45,28,31)(9,16,38,29,32)(10,17,39,22,33)(11,18,40,23,34)(12,19,41,24,35)(13,20,42,25,36), (1,19,15)(2,41,45)(3,35,31)(4,24,28)(5,12,8)(6,7,9)(10,13,11)(14,16,21)(17,20,18)(22,25,23)(26,27,29)(30,32,37)(33,36,34)(38,43,44)(39,42,40), (1,20,16)(2,42,38)(3,36,32)(4,25,29)(5,13,9)(6,12,11)(7,8,10)(14,15,17)(18,21,19)(22,27,28)(23,26,24)(30,31,33)(34,37,35)(39,44,45)(40,43,41), (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)>;
 
G:=Group( (1,2,4,3,5)(6,21,43,26,37)(7,14,44,27,30)(8,15,45,28,31)(9,16,38,29,32)(10,17,39,22,33)(11,18,40,23,34)(12,19,41,24,35)(13,20,42,25,36), (1,19,15)(2,41,45)(3,35,31)(4,24,28)(5,12,8)(6,7,9)(10,13,11)(14,16,21)(17,20,18)(22,25,23)(26,27,29)(30,32,37)(33,36,34)(38,43,44)(39,42,40), (1,20,16)(2,42,38)(3,36,32)(4,25,29)(5,13,9)(6,12,11)(7,8,10)(14,15,17)(18,21,19)(22,27,28)(23,26,24)(30,31,33)(34,37,35)(39,44,45)(40,43,41), (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) );
 
G=PermutationGroup([[(1,2,4,3,5),(6,21,43,26,37),(7,14,44,27,30),(8,15,45,28,31),(9,16,38,29,32),(10,17,39,22,33),(11,18,40,23,34),(12,19,41,24,35),(13,20,42,25,36)], [(1,19,15),(2,41,45),(3,35,31),(4,24,28),(5,12,8),(6,7,9),(10,13,11),(14,16,21),(17,20,18),(22,25,23),(26,27,29),(30,32,37),(33,36,34),(38,43,44),(39,42,40)], [(1,20,16),(2,42,38),(3,36,32),(4,25,29),(5,13,9),(6,12,11),(7,8,10),(14,15,17),(18,21,19),(22,27,28),(23,26,24),(30,31,33),(34,37,35),(39,44,45),(40,43,41)], [(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)]])
 

45 conjugacy classes

class 1  2  3 4A4B5A5B5C5D8A8B8C8D10A10B10C10D15A15B15C15D20A···20H40A···40P
order1234455558888101010101515151520···2040···40
size1989911119999999988889···99···9

45 irreducible representations

dim1111111188
type+++
imageC1C2C4C5C8C10C20C40F9C5×F9
kernelC5×F9C5×C32⋊C4C5×C3⋊S3F9C3×C15C32⋊C4C3⋊S3C32C5C1
# reps11244481614

Matrix representation of C5×F9 ►in GL8(𝔽241)

910000000
091000000
009100000
000910000
000091000
000009100
000000910
000000091
,
0010240000
0001240000
0000240001
0000240100
0000240010
0100240000
0000240000
1000240000
,
0000010240
0000001240
0100000240
0000000240
1000000240
0000100240
0010000240
0001000240
,
00100000
00000100
00010000
00000010
10000000
00000001
01000000
00001000

G:=sub<GL(8,GF(241))| [91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,91],[0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,240,240,240,240,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,240,240,240,240,240,240],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

C5×F9 in GAP, Magma, Sage, TeX 

C_5\times F_9
 
% in TeX
 
G:=Group("C5xF9");
 
// GroupNames label
 
G:=SmallGroup(360,123);
 
// by ID
 
G=gap.SmallGroup(360,123);
 
# by ID
 
G:=PCGroup([6,-2,-5,-2,-2,-3,3,60,50,3604,856,142,10085,1169,455]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
 
// generators/relations
 

Export

Subgroup lattice of C5×F9 in TeX

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