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## G = C3×F9order 216 = 23·33

### Direct product of C3 and F9

Aliases: C3×F9, C32⋊C24, C331C8, C3⋊S3.C12, C32⋊C4.1C6, (C3×C3⋊S3).1C4, (C3×C32⋊C4).1C2, SmallGroup(216,154)

Series: Derived Chief Lower central Upper central

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

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

Character table of C3×F9

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D 24E 24F 24G 24H size 1 9 1 1 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ3 ζ32 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 linear of order 3 ρ4 1 1 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ32 ζ3 -1 -1 -1 -1 ζ32 ζ3 ζ32 ζ3 ζ65 ζ65 ζ65 ζ65 ζ6 ζ6 ζ6 ζ6 linear of order 6 ρ5 1 1 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ32 ζ3 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 linear of order 3 ρ6 1 1 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ3 ζ32 -1 -1 -1 -1 ζ3 ζ32 ζ3 ζ32 ζ6 ζ6 ζ6 ζ6 ζ65 ζ65 ζ65 ζ65 linear of order 6 ρ7 1 1 1 1 1 1 1 -1 -1 1 1 i -i -i i -1 -1 -1 -1 -i -i i i -i -i i i linear of order 4 ρ8 1 1 1 1 1 1 1 -1 -1 1 1 -i i i -i -1 -1 -1 -1 i i -i -i i i -i -i linear of order 4 ρ9 1 -1 1 1 1 1 1 i -i -1 -1 ζ83 ζ85 ζ8 ζ87 i -i -i i ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 linear of order 8 ρ10 1 -1 1 1 1 1 1 -i i -1 -1 ζ85 ζ83 ζ87 ζ8 -i i i -i ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 linear of order 8 ρ11 1 -1 1 1 1 1 1 i -i -1 -1 ζ87 ζ8 ζ85 ζ83 i -i -i i ζ8 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ87 linear of order 8 ρ12 1 -1 1 1 1 1 1 -i i -1 -1 ζ8 ζ87 ζ83 ζ85 -i i i -i ζ87 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ8 linear of order 8 ρ13 1 1 ζ3 ζ32 1 ζ3 ζ32 -1 -1 ζ32 ζ3 -i i i -i ζ6 ζ65 ζ6 ζ65 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ43ζ3 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ43ζ32 linear of order 12 ρ14 1 1 ζ32 ζ3 1 ζ32 ζ3 -1 -1 ζ3 ζ32 i -i -i i ζ65 ζ6 ζ65 ζ6 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ4ζ32 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ4ζ3 linear of order 12 ρ15 1 1 ζ3 ζ32 1 ζ3 ζ32 -1 -1 ζ32 ζ3 i -i -i i ζ6 ζ65 ζ6 ζ65 ζ43ζ3 ζ43ζ3 ζ4ζ3 ζ4ζ3 ζ43ζ32 ζ43ζ32 ζ4ζ32 ζ4ζ32 linear of order 12 ρ16 1 1 ζ32 ζ3 1 ζ32 ζ3 -1 -1 ζ3 ζ32 -i i i -i ζ65 ζ6 ζ65 ζ6 ζ4ζ32 ζ4ζ32 ζ43ζ32 ζ43ζ32 ζ4ζ3 ζ4ζ3 ζ43ζ3 ζ43ζ3 linear of order 12 ρ17 1 -1 ζ32 ζ3 1 ζ32 ζ3 -i i ζ65 ζ6 ζ85 ζ83 ζ87 ζ8 ζ86ζ3 ζ82ζ32 ζ82ζ3 ζ86ζ32 ζ83ζ32 ζ87ζ32 ζ8ζ32 ζ85ζ32 ζ83ζ3 ζ87ζ3 ζ8ζ3 ζ85ζ3 linear of order 24 ρ18 1 -1 ζ3 ζ32 1 ζ3 ζ32 -i i ζ6 ζ65 ζ85 ζ83 ζ87 ζ8 ζ86ζ32 ζ82ζ3 ζ82ζ32 ζ86ζ3 ζ83ζ3 ζ87ζ3 ζ8ζ3 ζ85ζ3 ζ83ζ32 ζ87ζ32 ζ8ζ32 ζ85ζ32 linear of order 24 ρ19 1 -1 ζ3 ζ32 1 ζ3 ζ32 -i i ζ6 ζ65 ζ8 ζ87 ζ83 ζ85 ζ86ζ32 ζ82ζ3 ζ82ζ32 ζ86ζ3 ζ87ζ3 ζ83ζ3 ζ85ζ3 ζ8ζ3 ζ87ζ32 ζ83ζ32 ζ85ζ32 ζ8ζ32 linear of order 24 ρ20 1 -1 ζ32 ζ3 1 ζ32 ζ3 -i i ζ65 ζ6 ζ8 ζ87 ζ83 ζ85 ζ86ζ3 ζ82ζ32 ζ82ζ3 ζ86ζ32 ζ87ζ32 ζ83ζ32 ζ85ζ32 ζ8ζ32 ζ87ζ3 ζ83ζ3 ζ85ζ3 ζ8ζ3 linear of order 24 ρ21 1 -1 ζ32 ζ3 1 ζ32 ζ3 i -i ζ65 ζ6 ζ87 ζ8 ζ85 ζ83 ζ82ζ3 ζ86ζ32 ζ86ζ3 ζ82ζ32 ζ8ζ32 ζ85ζ32 ζ83ζ32 ζ87ζ32 ζ8ζ3 ζ85ζ3 ζ83ζ3 ζ87ζ3 linear of order 24 ρ22 1 -1 ζ3 ζ32 1 ζ3 ζ32 i -i ζ6 ζ65 ζ83 ζ85 ζ8 ζ87 ζ82ζ32 ζ86ζ3 ζ86ζ32 ζ82ζ3 ζ85ζ3 ζ8ζ3 ζ87ζ3 ζ83ζ3 ζ85ζ32 ζ8ζ32 ζ87ζ32 ζ83ζ32 linear of order 24 ρ23 1 -1 ζ3 ζ32 1 ζ3 ζ32 i -i ζ6 ζ65 ζ87 ζ8 ζ85 ζ83 ζ82ζ32 ζ86ζ3 ζ86ζ32 ζ82ζ3 ζ8ζ3 ζ85ζ3 ζ83ζ3 ζ87ζ3 ζ8ζ32 ζ85ζ32 ζ83ζ32 ζ87ζ32 linear of order 24 ρ24 1 -1 ζ32 ζ3 1 ζ32 ζ3 i -i ζ65 ζ6 ζ83 ζ85 ζ8 ζ87 ζ82ζ3 ζ86ζ32 ζ86ζ3 ζ82ζ32 ζ85ζ32 ζ8ζ32 ζ87ζ32 ζ83ζ32 ζ85ζ3 ζ8ζ3 ζ87ζ3 ζ83ζ3 linear of order 24 ρ25 8 0 8 8 -1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F9 ρ26 8 0 -4-4√-3 -4+4√-3 -1 ζ6 ζ65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful ρ27 8 0 -4+4√-3 -4-4√-3 -1 ζ65 ζ6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 complex faithful

Permutation representations of C3×F9
On 24 points - transitive group 24T567
Generators in S24
(1 12 21)(2 13 22)(3 14 23)(4 15 24)(5 16 17)(6 9 18)(7 10 19)(8 11 20)
(2 13 22)(3 14 23)(4 24 15)(6 18 9)(7 19 10)(8 11 20)
(1 12 21)(3 14 23)(4 15 24)(5 17 16)(7 19 10)(8 20 11)
(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)| (1,12,21)(2,13,22)(3,14,23)(4,15,24)(5,16,17)(6,9,18)(7,10,19)(8,11,20), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (1,12,21)(3,14,23)(4,15,24)(5,17,16)(7,19,10)(8,20,11), (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( (1,12,21)(2,13,22)(3,14,23)(4,15,24)(5,16,17)(6,9,18)(7,10,19)(8,11,20), (2,13,22)(3,14,23)(4,24,15)(6,18,9)(7,19,10)(8,11,20), (1,12,21)(3,14,23)(4,15,24)(5,17,16)(7,19,10)(8,20,11), (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([[(1,12,21),(2,13,22),(3,14,23),(4,15,24),(5,16,17),(6,9,18),(7,10,19),(8,11,20)], [(2,13,22),(3,14,23),(4,24,15),(6,18,9),(7,19,10),(8,11,20)], [(1,12,21),(3,14,23),(4,15,24),(5,17,16),(7,19,10),(8,20,11)], [(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(24,567);

On 27 points - transitive group 27T78
Generators in S27
(1 3 2)(4 19 23)(5 12 24)(6 13 25)(7 14 26)(8 15 27)(9 16 20)(10 17 21)(11 18 22)
(1 15 19)(2 8 4)(3 27 23)(5 10 11)(6 9 7)(12 17 18)(13 16 14)(20 26 25)(21 22 24)
(1 16 12)(2 9 5)(3 20 24)(4 6 11)(7 10 8)(13 18 19)(14 17 15)(21 27 26)(22 23 25)
(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)(4,19,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,20)(10,17,21)(11,18,22), (1,15,19)(2,8,4)(3,27,23)(5,10,11)(6,9,7)(12,17,18)(13,16,14)(20,26,25)(21,22,24), (1,16,12)(2,9,5)(3,20,24)(4,6,11)(7,10,8)(13,18,19)(14,17,15)(21,27,26)(22,23,25), (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)(4,19,23)(5,12,24)(6,13,25)(7,14,26)(8,15,27)(9,16,20)(10,17,21)(11,18,22), (1,15,19)(2,8,4)(3,27,23)(5,10,11)(6,9,7)(12,17,18)(13,16,14)(20,26,25)(21,22,24), (1,16,12)(2,9,5)(3,20,24)(4,6,11)(7,10,8)(13,18,19)(14,17,15)(21,27,26)(22,23,25), (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),(4,19,23),(5,12,24),(6,13,25),(7,14,26),(8,15,27),(9,16,20),(10,17,21),(11,18,22)], [(1,15,19),(2,8,4),(3,27,23),(5,10,11),(6,9,7),(12,17,18),(13,16,14),(20,26,25),(21,22,24)], [(1,16,12),(2,9,5),(3,20,24),(4,6,11),(7,10,8),(13,18,19),(14,17,15),(21,27,26),(22,23,25)], [(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(27,78);

C3×F9 is a maximal subgroup of   F9⋊S3

Matrix representation of C3×F9 in GL8(𝔽73)

 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 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 1 0 8 0 0 0 0 0 9 0 0 64 0 0 0 0 64 0 0 0 64 0 0 0 72 0 0 0 0 8 0 0 72 0 0 0 0 0 8 0 64 0 0 0 0 0 0 64
,
 64 0 0 0 0 0 0 0 65 8 0 0 0 0 0 0 65 0 8 0 0 0 0 0 0 0 0 64 0 0 0 0 8 0 0 0 8 0 0 0 0 0 0 0 0 64 0 0 9 0 0 0 0 0 1 0 9 0 0 0 0 0 0 1
,
 72 0 0 0 7 0 0 0 0 0 0 0 72 1 0 0 0 0 0 0 72 0 1 0 0 0 0 0 72 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0

G:=sub<GL(8,GF(73))| [64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64],[1,0,1,9,64,72,72,64,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64],[64,65,65,0,8,0,9,9,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,7,72,72,72,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C3×F9 in GAP, Magma, Sage, TeX

C_3\times F_9
% in TeX

G:=Group("C3xF9");
// GroupNames label

G:=SmallGroup(216,154);
// by ID

G=gap.SmallGroup(216,154);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,-3,3,36,50,2164,856,142,6053,1169,455]);
// Polycyclic

G:=Group<a,b,c,d|a^3=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

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