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G = C5⋊2F9  order 360 = 23·32·5

The semidirect product of C5 and F9 acting via F9/C32⋊C4=C2

metabelian, soluble, monomial, A-group

Aliases: C5⋊2F9, (C3×C15)⋊2C8, C32⋊(C5⋊2C8), C3⋊S3.Dic5, C32⋊C4.1D5, (C5×C3⋊S3).2C4, (C5×C32⋊C4).3C2, SmallGroup(360,124)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C15 — C5⋊2F9
Chief series
C1 — C5 — C3×C15 — C5×C3⋊S3 — C5×C32⋊C4 — C5⋊2F9
Lower central
C3×C15 — C5⋊2F9
Upper central
C1

Generators and relations for C5⋊2F9
 G = < a,b,c,d | a5=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >

C1
9C2
4C3
C5
9C4
12S3
C32
9C10
4C15
45C8
C3⋊S3
9C20
12C5×S3
C3×C15
C32⋊C4
9C5⋊2C8
C5×C3⋊S3
5F9
C5×C32⋊C4
C5⋊2F9

Character table of C5⋊2F9

 class 1234A4B5A5B8A8B8C8D10A10B15A15B15C15D20A20B20C20D
 size 1989922454545451818888818181818
ρ1111111111111111111111    trivial
ρ21111111-1-1-1-11111111111    linear of order 2
ρ3111-1-111i-ii-i111111-1-1-1-1    linear of order 4
ρ4111-1-111-ii-ii111111-1-1-1-1    linear of order 4
ρ51-11i-i11ζ87ζ85ζ83ζ8-1-11111ii-i-i    linear of order 8
ρ61-11-ii11ζ85ζ87ζ8ζ83-1-11111-i-iii    linear of order 8
ρ71-11i-i11ζ83ζ8ζ87ζ85-1-11111ii-i-i    linear of order 8
ρ81-11-ii11ζ8ζ83ζ85ζ87-1-11111-i-iii    linear of order 8
ρ922222-1+√5/2-1-√5/20000-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ1022222-1-√5/2-1+√5/20000-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ11222-2-2-1+√5/2-1-√5/20000-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/21+√5/21-√5/21-√5/21+√5/2    symplectic lifted from Dic5, Schur index 2
ρ12222-2-2-1-√5/2-1+√5/20000-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/21-√5/21+√5/21+√5/21-√5/2    symplectic lifted from Dic5, Schur index 2
ρ132-222i-2i-1-√5/2-1+√5/200001-√5/21+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2ζ4ζ54+ζ4ζ5ζ4ζ53+ζ4ζ52ζ43ζ53+ζ43ζ52ζ43ζ54+ζ43ζ5    complex lifted from C5⋊2C8, Schur index 2
ρ142-222i-2i-1+√5/2-1-√5/200001+√5/21-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2ζ4ζ53+ζ4ζ52ζ4ζ54+ζ4ζ5ζ43ζ54+ζ43ζ5ζ43ζ53+ζ43ζ52    complex lifted from C5⋊2C8, Schur index 2
ρ152-22-2i2i-1-√5/2-1+√5/200001-√5/21+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2ζ43ζ54+ζ43ζ5ζ43ζ53+ζ43ζ52ζ4ζ53+ζ4ζ52ζ4ζ54+ζ4ζ5    complex lifted from C5⋊2C8, Schur index 2
ρ162-22-2i2i-1+√5/2-1-√5/200001+√5/21-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2ζ43ζ53+ζ43ζ52ζ43ζ54+ζ43ζ5ζ4ζ54+ζ4ζ5ζ4ζ53+ζ4ζ52    complex lifted from C5⋊2C8, Schur index 2
ρ1780-10088000000-1-1-1-10000    orthogonal lifted from F9
ρ1880-100-2+2√5-2-2√5000000ζ53-2ζ52ζ54-2ζ5-2ζ53+ζ52-2ζ54+ζ50000    complex faithful
ρ1980-100-2-2√5-2+2√5000000-2ζ54+ζ5ζ53-2ζ52ζ54-2ζ5-2ζ53+ζ520000    complex faithful
ρ2080-100-2+2√5-2-2√5000000-2ζ53+ζ52-2ζ54+ζ5ζ53-2ζ52ζ54-2ζ50000    complex faithful
ρ2180-100-2-2√5-2+2√5000000ζ54-2ζ5-2ζ53+ζ52-2ζ54+ζ5ζ53-2ζ520000    complex faithful

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

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

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

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

Matrix representation of C5⋊2F9 ►in GL10(𝔽241)

98000000000
749100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
001019200000
000024000000
000124000000
000024000100
000024010000
000024001000
00192240239240240240240240
000024000010
,
1000000000
0100000000
001001920000
000002401000
000002400100
000002400010
000002400001
00192240240239240240240240
000002400000
000102400000
,
2714400000000
6421400000000
0010000000
0000000001
0000000100
0000100000
0000000010
0000001000
0001000000
00192240240240240240240240

G:=sub<GL(10,GF(241))| [98,74,0,0,0,0,0,0,0,0,0,91,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,192,0,0,0,0,0,1,0,0,0,240,0,0,0,192,240,240,240,240,240,239,240,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,0,240,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,192,0,0,0,0,0,0,0,0,0,240,0,1,0,0,0,0,0,0,0,240,0,0,0,0,192,240,240,240,240,239,240,240,0,0,0,1,0,0,0,240,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,0,1,0,240,0,0,0,0,0,0,0,0,1,240,0,0],[27,64,0,0,0,0,0,0,0,0,144,214,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,192,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,1,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,240,0,0,0,0,1,0,0,0,0,240,0,0,0,0,0,0,1,0,0,240,0,0,0,1,0,0,0,0,0,240] >;
 

C5⋊2F9 in GAP, Magma, Sage, TeX 

C_5\rtimes_2F_9
 
% in TeX
 
G:=Group("C5:2F9");
 
// GroupNames label
 
G:=SmallGroup(360,124);
 
// by ID
 
G=gap.SmallGroup(360,124);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-5,12,31,771,489,111,244,490,376,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
 
// generators/relations
 

Export

Subgroup lattice of C5⋊2F9 in TeX
Character table of C5⋊2F9 in TeX

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