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G = C2×F9  order 144 = 24·32

Direct product of C2 and F9

direct product, metabelian, soluble, monomial, A-group

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 >

C1
C2
9C2
9C2
4C3
9C22
9C4
9C4
4C6
12S3
12S3
C32
9C8
9C8
9C2×C4
12D6
C3×C6
C3⋊S3
C3⋊S3
9C2×C8
C32⋊C4
C2×C3⋊S3
C32⋊C4
C2×C32⋊C4
F9
F9
C2×F9

Character table of C2×F9

 class 12A2B2C34A4B4C4D68A8B8C8D8E8F8G8H
 size 119989999899999999
ρ1111111111111111111    trivial
ρ21-11-11-11-11-1-1111-1-1-11    linear of order 2
ρ31-11-11-11-11-11-1-1-1111-1    linear of order 2
ρ41111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511111-1-1-1-11-ii-i-iii-ii    linear of order 4
ρ611111-1-1-1-11i-iii-i-ii-i    linear of order 4
ρ71-11-111-11-1-1ii-i-i-i-iii    linear of order 4
ρ81-11-111-11-1-1-i-iiiii-i-i    linear of order 4
ρ91-1-111-iii-i-1ζ85ζ83ζ85ζ8ζ83ζ87ζ8ζ87    linear of order 8
ρ101-1-111-iii-i-1ζ8ζ87ζ8ζ85ζ87ζ83ζ85ζ83    linear of order 8
ρ1111-1-11-i-iii1ζ83ζ8ζ87ζ83ζ85ζ8ζ87ζ85    linear of order 8
ρ1211-1-11-i-iii1ζ87ζ85ζ83ζ87ζ8ζ85ζ83ζ8    linear of order 8
ρ131-1-111i-i-ii-1ζ87ζ8ζ87ζ83ζ8ζ85ζ83ζ85    linear of order 8
ρ1411-1-11ii-i-i1ζ8ζ83ζ85ζ8ζ87ζ83ζ85ζ87    linear of order 8
ρ151-1-111i-i-ii-1ζ83ζ85ζ83ζ87ζ85ζ8ζ87ζ8    linear of order 8
ρ1611-1-11ii-i-i1ζ85ζ87ζ8ζ85ζ83ζ87ζ8ζ83    linear of order 8
ρ178-800-10000100000000    orthogonal faithful
ρ188800-10000-100000000    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)

-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
00000010
00000001
-1-1-1-1-1-1-1-1
10000000
01000000
00100000
00010000
00001000
,
00100000
10000000
01000000
00000100
00010000
00001000
-1-1-1-1-1-1-1-1
00000010
,
10000000
00010000
00000010
00001000
00000001
01000000
-1-1-1-1-1-1-1-1
00100000

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
 

Export

Subgroup lattice of C2×F9 in TeX
Character table of C2×F9 in TeX

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