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G = C3⋊F9  order 216 = 23·33

The semidirect product of C3 and F9 acting via F9/C32⋊C4=C2

metabelian, soluble, monomial, A-group

Aliases: C3⋊F9, C33⋊2C8, C32⋊(C3⋊C8), C3⋊S3.Dic3, C32⋊C4.1S3, (C3×C3⋊S3).2C4, (C3×C32⋊C4).3C2, SmallGroup(216,155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C33 — C3⋊F9
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C3×C32⋊C4 — C3⋊F9
Lower central
C33 — C3⋊F9
Upper central
C1

Generators and relations for C3⋊F9
 G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >

C1
9C2
C3
4C3
8C3
9C4
9C6
12S3
C32
4C32
8C32
27C8
9C12
C3⋊S3
12C3×S3
C33
9C3⋊C8
C32⋊C4
C3×C3⋊S3
3F9
C3×C32⋊C4
C3⋊F9

Character table of C3⋊F9

 class 123A3B3C3D4A4B68A8B8C8D12A12B
 size 1928889918272727271818
ρ1111111111111111    trivial
ρ2111111111-1-1-1-111    linear of order 2
ρ3111111-1-11i-i-ii-1-1    linear of order 4
ρ4111111-1-11-iii-i-1-1    linear of order 4
ρ51-11111-ii-1ζ8ζ87ζ83ζ85i-i    linear of order 8
ρ61-11111-ii-1ζ85ζ83ζ87ζ8i-i    linear of order 8
ρ71-11111i-i-1ζ87ζ8ζ85ζ83-ii    linear of order 8
ρ81-11111i-i-1ζ83ζ85ζ8ζ87-ii    linear of order 8
ρ922-1-1-1222-10000-1-1    orthogonal lifted from S3
ρ1022-1-1-12-2-2-1000011    symplectic lifted from Dic3, Schur index 2
ρ112-2-1-1-12-2i2i10000-ii    complex lifted from C3⋊C8
ρ122-2-1-1-122i-2i10000i-i    complex lifted from C3⋊C8
ρ13808-1-1-1000000000    orthogonal lifted from F9
ρ1480-41-3√-3/21+3√-3/2-1000000000    complex faithful
ρ1580-41+3√-3/21-3√-3/2-1000000000    complex faithful

Permutation representations of C3⋊F9
►On 24 points - transitive group 24T568
Generators in S24
(1 22 14)(2 15 23)(3 24 16)(4 9 17)(5 18 10)(6 11 19)(7 20 12)(8 13 21)

(2 15 23)(3 16 24)(4 17 9)(6 19 11)(7 20 12)(8 13 21)

(1 14 22)(3 16 24)(4 9 17)(5 18 10)(7 20 12)(8 21 13)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
 
G:=sub<Sym(24)| (1,22,14)(2,15,23)(3,24,16)(4,9,17)(5,18,10)(6,11,19)(7,20,12)(8,13,21), (2,15,23)(3,16,24)(4,17,9)(6,19,11)(7,20,12)(8,13,21), (1,14,22)(3,16,24)(4,9,17)(5,18,10)(7,20,12)(8,21,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)>;
 
G:=Group( (1,22,14)(2,15,23)(3,24,16)(4,9,17)(5,18,10)(6,11,19)(7,20,12)(8,13,21), (2,15,23)(3,16,24)(4,17,9)(6,19,11)(7,20,12)(8,13,21), (1,14,22)(3,16,24)(4,9,17)(5,18,10)(7,20,12)(8,21,13), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24) );
 
G=PermutationGroup([[(1,22,14),(2,15,23),(3,24,16),(4,9,17),(5,18,10),(6,11,19),(7,20,12),(8,13,21)], [(2,15,23),(3,16,24),(4,17,9),(6,19,11),(7,20,12),(8,13,21)], [(1,14,22),(3,16,24),(4,9,17),(5,18,10),(7,20,12),(8,21,13)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)]])
 
G:=TransitiveGroup(24,568);
 

►On 27 points - transitive group 27T80
Generators in S27
(1 3 2)(4 24 14)(5 15 25)(6 26 16)(7 17 27)(8 20 18)(9 19 21)(10 22 12)(11 13 23)

(1 17 13)(2 7 11)(3 27 23)(4 21 10)(5 20 26)(6 25 8)(9 22 24)(12 14 19)(15 18 16)

(1 18 14)(2 20 24)(3 8 4)(5 22 11)(6 21 27)(7 26 9)(10 23 25)(12 13 15)(16 19 17)

(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)
 
G:=sub<Sym(27)| (1,3,2)(4,24,14)(5,15,25)(6,26,16)(7,17,27)(8,20,18)(9,19,21)(10,22,12)(11,13,23), (1,17,13)(2,7,11)(3,27,23)(4,21,10)(5,20,26)(6,25,8)(9,22,24)(12,14,19)(15,18,16), (1,18,14)(2,20,24)(3,8,4)(5,22,11)(6,21,27)(7,26,9)(10,23,25)(12,13,15)(16,19,17), (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)>;
 
G:=Group( (1,3,2)(4,24,14)(5,15,25)(6,26,16)(7,17,27)(8,20,18)(9,19,21)(10,22,12)(11,13,23), (1,17,13)(2,7,11)(3,27,23)(4,21,10)(5,20,26)(6,25,8)(9,22,24)(12,14,19)(15,18,16), (1,18,14)(2,20,24)(3,8,4)(5,22,11)(6,21,27)(7,26,9)(10,23,25)(12,13,15)(16,19,17), (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) );
 
G=PermutationGroup([[(1,3,2),(4,24,14),(5,15,25),(6,26,16),(7,17,27),(8,20,18),(9,19,21),(10,22,12),(11,13,23)], [(1,17,13),(2,7,11),(3,27,23),(4,21,10),(5,20,26),(6,25,8),(9,22,24),(12,14,19),(15,18,16)], [(1,18,14),(2,20,24),(3,8,4),(5,22,11),(6,21,27),(7,26,9),(10,23,25),(12,13,15),(16,19,17)], [(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)]])
 
G:=TransitiveGroup(27,80);
 

C3⋊F9 is a maximal subgroup of   S3×F9  C33⋊SD16  C33⋊3SD16
C3⋊F9 is a maximal quotient of   C6.F9

Matrix representation of C3⋊F9 ►in GL8(𝔽73)

80000000
08000000
00800000
00080000
000064000
000006400
000000640
000000064
,
10000000
01000000
00800000
000640000
00008000
000006400
000000640
00000008
,
640000000
08000000
00800000
000640000
000064000
00000800
00000010
00000001
,
00000100
00001000
00000001
00000010
00010000
00100000
10000000
01000000

G:=sub<GL(8,GF(73))| [8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8],[64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0] >;
 

C3⋊F9 in GAP, Magma, Sage, TeX 

C_3\rtimes F_9
 
% in TeX
 
G:=Group("C3:F9");
 
// GroupNames label
 
G:=SmallGroup(216,155);
 
// by ID
 
G=gap.SmallGroup(216,155);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-3,12,31,771,489,111,244,490,376,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=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 C3⋊F9 in TeX
Character table of C3⋊F9 in TeX

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