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## G = C2×F9order 144 = 24·32

### Direct product of C2 and F9

Aliases: C2×F9, C3⋊S3⋊C8, (C3×C6)⋊C8, C32⋊(C2×C8), C32⋊C4.C4, C32⋊C4.2C22, (C2×C3⋊S3).C4, C3⋊S3.1(C2×C4), (C2×C32⋊C4).4C2, SmallGroup(144,185)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×F9
 Chief series C1 — C32 — C3⋊S3 — C32⋊C4 — F9 — C2×F9
 Lower central C32 — C2×F9
 Upper central C1 — C2

Generators and relations for C2×F9
G = < a,b,c,d | a2=b3=c3=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

Character table of C2×F9

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 9 9 8 9 9 9 9 8 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 trivial ρ2 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 1 -1 1 -1 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 -1 -1 1 -i i -i -i i i -i i linear of order 4 ρ6 1 1 1 1 1 -1 -1 -1 -1 1 i -i i i -i -i i -i linear of order 4 ρ7 1 -1 1 -1 1 1 -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 -i -i i i i i -i -i linear of order 4 ρ9 1 -1 -1 1 1 -i i i -i -1 ζ85 ζ83 ζ85 ζ8 ζ83 ζ87 ζ8 ζ87 linear of order 8 ρ10 1 -1 -1 1 1 -i i i -i -1 ζ8 ζ87 ζ8 ζ85 ζ87 ζ83 ζ85 ζ83 linear of order 8 ρ11 1 1 -1 -1 1 -i -i i i 1 ζ83 ζ8 ζ87 ζ83 ζ85 ζ8 ζ87 ζ85 linear of order 8 ρ12 1 1 -1 -1 1 -i -i i i 1 ζ87 ζ85 ζ83 ζ87 ζ8 ζ85 ζ83 ζ8 linear of order 8 ρ13 1 -1 -1 1 1 i -i -i i -1 ζ87 ζ8 ζ87 ζ83 ζ8 ζ85 ζ83 ζ85 linear of order 8 ρ14 1 1 -1 -1 1 i i -i -i 1 ζ8 ζ83 ζ85 ζ8 ζ87 ζ83 ζ85 ζ87 linear of order 8 ρ15 1 -1 -1 1 1 i -i -i i -1 ζ83 ζ85 ζ83 ζ87 ζ85 ζ8 ζ87 ζ8 linear of order 8 ρ16 1 1 -1 -1 1 i i -i -i 1 ζ85 ζ87 ζ8 ζ85 ζ83 ζ87 ζ8 ζ83 linear of order 8 ρ17 8 -8 0 0 -1 0 0 0 0 1 0 0 0 0 0 0 0 0 orthogonal faithful ρ18 8 8 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 orthogonal lifted from F9

Permutation representations of C2×F9
On 18 points - transitive group 18T59
Generators in S18
(1 2)(3 17)(4 18)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)
(1 18 14)(2 4 8)(3 16 5)(6 13 15)(7 9 12)(10 11 17)
(1 5 9)(2 11 15)(3 12 18)(4 17 6)(7 14 16)(8 10 13)
(1 2)(3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18)

G:=sub<Sym(18)| (1,2)(3,17)(4,18)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16), (1,18,14)(2,4,8)(3,16,5)(6,13,15)(7,9,12)(10,11,17), (1,5,9)(2,11,15)(3,12,18)(4,17,6)(7,14,16)(8,10,13), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18)>;

G:=Group( (1,2)(3,17)(4,18)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16), (1,18,14)(2,4,8)(3,16,5)(6,13,15)(7,9,12)(10,11,17), (1,5,9)(2,11,15)(3,12,18)(4,17,6)(7,14,16)(8,10,13), (1,2)(3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18) );

G=PermutationGroup([[(1,2),(3,17),(4,18),(5,11),(6,12),(7,13),(8,14),(9,15),(10,16)], [(1,18,14),(2,4,8),(3,16,5),(6,13,15),(7,9,12),(10,11,17)], [(1,5,9),(2,11,15),(3,12,18),(4,17,6),(7,14,16),(8,10,13)], [(1,2),(3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18)]])

G:=TransitiveGroup(18,59);

On 24 points - transitive group 24T255
Generators in S24
(1 6)(2 7)(3 8)(4 5)(9 21)(10 22)(11 23)(12 24)(13 17)(14 18)(15 19)(16 20)
(1 23 19)(2 20 24)(4 22 18)(5 10 14)(6 11 15)(7 16 12)
(1 23 19)(2 24 20)(3 21 17)(6 11 15)(7 12 16)(8 9 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)| (1,6)(2,7)(3,8)(4,5)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,23,19)(2,20,24)(4,22,18)(5,10,14)(6,11,15)(7,16,12), (1,23,19)(2,24,20)(3,21,17)(6,11,15)(7,12,16)(8,9,13), (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( (1,6)(2,7)(3,8)(4,5)(9,21)(10,22)(11,23)(12,24)(13,17)(14,18)(15,19)(16,20), (1,23,19)(2,20,24)(4,22,18)(5,10,14)(6,11,15)(7,16,12), (1,23,19)(2,24,20)(3,21,17)(6,11,15)(7,12,16)(8,9,13), (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([[(1,6),(2,7),(3,8),(4,5),(9,21),(10,22),(11,23),(12,24),(13,17),(14,18),(15,19),(16,20)], [(1,23,19),(2,20,24),(4,22,18),(5,10,14),(6,11,15),(7,16,12)], [(1,23,19),(2,24,20),(3,21,17),(6,11,15),(7,12,16),(8,9,13)], [(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(24,255);

On 24 points - transitive group 24T256
Generators in S24
(1 5)(2 6)(3 7)(4 8)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(2 23 12)(3 24 13)(4 14 17)(6 16 19)(7 9 20)(8 21 10)
(1 22 11)(3 24 13)(4 17 14)(5 15 18)(7 9 20)(8 10 21)
(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)| (1,5)(2,6)(3,7)(4,8)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (2,23,12)(3,24,13)(4,14,17)(6,16,19)(7,9,20)(8,21,10), (1,22,11)(3,24,13)(4,17,14)(5,15,18)(7,9,20)(8,10,21), (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( (1,5)(2,6)(3,7)(4,8)(9,24)(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23), (2,23,12)(3,24,13)(4,14,17)(6,16,19)(7,9,20)(8,21,10), (1,22,11)(3,24,13)(4,17,14)(5,15,18)(7,9,20)(8,10,21), (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([[(1,5),(2,6),(3,7),(4,8),(9,24),(10,17),(11,18),(12,19),(13,20),(14,21),(15,22),(16,23)], [(2,23,12),(3,24,13),(4,14,17),(6,16,19),(7,9,20),(8,21,10)], [(1,22,11),(3,24,13),(4,17,14),(5,15,18),(7,9,20),(8,10,21)], [(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(24,256);

C2×F9 is a maximal subgroup of   C2.AΓL1(𝔽9)  PSU3(𝔽2)⋊C4  F9⋊C4  C4⋊F9  C22⋊F9
C2×F9 is a maximal quotient of   C4.3F9  C4.F9  C4⋊F9  C22.F9  C22⋊F9

Matrix representation of C2×F9 in GL8(ℤ)

 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 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 -1 -1 -1 -1 -1 -1 -1 -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 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 0 0 0 1 0 0 0 0 0 0 0 0 1 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 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 0 0 1 0 0 0 0 0

G:=sub<GL(8,Integers())| [-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0],[0,1,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0],[1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,1,0,1,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0,0,0,-1,0,0,0,0,0,1,0,-1,0] >;

C2×F9 in GAP, Magma, Sage, TeX

C_2\times F_9
% in TeX

G:=Group("C2xF9");
// GroupNames label

G:=SmallGroup(144,185);
// by ID

G=gap.SmallGroup(144,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,24,50,1444,856,142,4037,1169,455]);
// Polycyclic

G:=Group<a,b,c,d|a^2=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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