Copied to
clipboard

G = C3⋊F9order 216 = 23·33

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

metabelian, soluble, monomial, A-group

Aliases: C3⋊F9, C332C8, 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

C1C33 — C3⋊F9
C1C3C33C3×C3⋊S3C3×C32⋊C4 — C3⋊F9
C33 — C3⋊F9
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 >

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

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 11)(2 12 23)(3 24 13)(4 14 17)(5 18 15)(6 16 19)(7 20 9)(8 10 21)
(2 12 23)(3 13 24)(4 17 14)(6 19 16)(7 20 9)(8 10 21)
(1 11 22)(3 13 24)(4 14 17)(5 18 15)(7 20 9)(8 21 10)
(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)| (1,22,11)(2,12,23)(3,24,13)(4,14,17)(5,18,15)(6,16,19)(7,20,9)(8,10,21), (2,12,23)(3,13,24)(4,17,14)(6,19,16)(7,20,9)(8,10,21), (1,11,22)(3,13,24)(4,14,17)(5,18,15)(7,20,9)(8,21,10), (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( (1,22,11)(2,12,23)(3,24,13)(4,14,17)(5,18,15)(6,16,19)(7,20,9)(8,10,21), (2,12,23)(3,13,24)(4,17,14)(6,19,16)(7,20,9)(8,10,21), (1,11,22)(3,13,24)(4,14,17)(5,18,15)(7,20,9)(8,21,10), (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([(1,22,11),(2,12,23),(3,24,13),(4,14,17),(5,18,15),(6,16,19),(7,20,9),(8,10,21)], [(2,12,23),(3,13,24),(4,17,14),(6,19,16),(7,20,9),(8,10,21)], [(1,11,22),(3,13,24),(4,14,17),(5,18,15),(7,20,9),(8,21,10)], [(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 27 16)(5 17 20)(6 21 18)(7 19 22)(8 23 12)(9 13 24)(10 25 14)(11 15 26)
(1 11 7)(2 26 22)(3 15 19)(4 10 9)(5 6 8)(12 20 18)(13 27 25)(14 24 16)(17 21 23)
(1 4 8)(2 16 12)(3 27 23)(5 11 10)(6 7 9)(13 21 19)(14 20 26)(15 25 17)(18 22 24)
(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)(4,27,16)(5,17,20)(6,21,18)(7,19,22)(8,23,12)(9,13,24)(10,25,14)(11,15,26), (1,11,7)(2,26,22)(3,15,19)(4,10,9)(5,6,8)(12,20,18)(13,27,25)(14,24,16)(17,21,23), (1,4,8)(2,16,12)(3,27,23)(5,11,10)(6,7,9)(13,21,19)(14,20,26)(15,25,17)(18,22,24), (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)(4,27,16)(5,17,20)(6,21,18)(7,19,22)(8,23,12)(9,13,24)(10,25,14)(11,15,26), (1,11,7)(2,26,22)(3,15,19)(4,10,9)(5,6,8)(12,20,18)(13,27,25)(14,24,16)(17,21,23), (1,4,8)(2,16,12)(3,27,23)(5,11,10)(6,7,9)(13,21,19)(14,20,26)(15,25,17)(18,22,24), (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),(4,27,16),(5,17,20),(6,21,18),(7,19,22),(8,23,12),(9,13,24),(10,25,14),(11,15,26)], [(1,11,7),(2,26,22),(3,15,19),(4,10,9),(5,6,8),(12,20,18),(13,27,25),(14,24,16),(17,21,23)], [(1,4,8),(2,16,12),(3,27,23),(5,11,10),(6,7,9),(13,21,19),(14,20,26),(15,25,17),(18,22,24)], [(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  C333SD16
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

׿
×
𝔽