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G = C2×C3.A4  order 72 = 23·32

Direct product of C2 and C3.A4

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

Aliases: C2×C3.A4, C23⋊C9, C22⋊C18, C6.2A4, C3.(C2×A4), (C2×C6).C6, (C22×C6).C3, SmallGroup(72,16)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C2×C3.A4
Chief series
C1 — C22 — C2×C6 — C3.A4 — C2×C3.A4
Lower central
C22 — C2×C3.A4
Upper central
C1 — C6

Generators and relations for C2×C3.A4
 G = < a,b,c,d,e | a2=b3=c2=d2=1, e3=b, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

C1
C2
3C2
3C2
C3
C22
3C22
3C22
C6
3C6
3C6
4C9
C23
C2×C6
3C2×C6
3C2×C6
4C18
C22×C6
C3.A4
C2×C3.A4

Character table of C2×C3.A4

 class 12A2B2C3A3B6A6B6C6D6E6F9A9B9C9D9E9F18A18B18C18D18E18F
 size 113311113333444444444444
ρ1111111111111111111111111    trivial
ρ21-1-1111-1-111-1-1111111-1-1-1-1-1-1    linear of order 2
ρ3111111111111ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ4111111111111ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ51-1-1111-1-111-1-1ζ3ζ3ζ32ζ32ζ32ζ3ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ61-1-1111-1-111-1-1ζ32ζ32ζ3ζ3ζ3ζ32ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ71111ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ9ζ97ζ92ζ95ζ98ζ94ζ9ζ94ζ97ζ92ζ95ζ98    linear of order 9
ρ81-1-11ζ32ζ3ζ65ζ6ζ3ζ32ζ6ζ65ζ95ζ98ζ9ζ97ζ94ζ92-ζ95-ζ92-ζ98-ζ9-ζ97-ζ94    linear of order 18
ρ91111ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ94ζ9ζ98ζ92ζ95ζ97ζ94ζ97ζ9ζ98ζ92ζ95    linear of order 9
ρ101-1-11ζ3ζ32ζ6ζ65ζ32ζ3ζ65ζ6ζ9ζ97ζ92ζ95ζ98ζ94-ζ9-ζ94-ζ97-ζ92-ζ95-ζ98    linear of order 18
ρ111-1-11ζ3ζ32ζ6ζ65ζ32ζ3ζ65ζ6ζ97ζ94ζ95ζ98ζ92ζ9-ζ97-ζ9-ζ94-ζ95-ζ98-ζ92    linear of order 18
ρ121111ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ95ζ98ζ9ζ97ζ94ζ92ζ95ζ92ζ98ζ9ζ97ζ94    linear of order 9
ρ131-1-11ζ3ζ32ζ6ζ65ζ32ζ3ζ65ζ6ζ94ζ9ζ98ζ92ζ95ζ97-ζ94-ζ97-ζ9-ζ98-ζ92-ζ95    linear of order 18
ρ141-1-11ζ32ζ3ζ65ζ6ζ3ζ32ζ6ζ65ζ92ζ95ζ94ζ9ζ97ζ98-ζ92-ζ98-ζ95-ζ94-ζ9-ζ97    linear of order 18
ρ151111ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ92ζ95ζ94ζ9ζ97ζ98ζ92ζ98ζ95ζ94ζ9ζ97    linear of order 9
ρ161111ζ32ζ3ζ3ζ32ζ3ζ32ζ32ζ3ζ98ζ92ζ97ζ94ζ9ζ95ζ98ζ95ζ92ζ97ζ94ζ9    linear of order 9
ρ171-1-11ζ32ζ3ζ65ζ6ζ3ζ32ζ6ζ65ζ98ζ92ζ97ζ94ζ9ζ95-ζ98-ζ95-ζ92-ζ97-ζ94-ζ9    linear of order 18
ρ181111ζ3ζ32ζ32ζ3ζ32ζ3ζ3ζ32ζ97ζ94ζ95ζ98ζ92ζ9ζ97ζ9ζ94ζ95ζ98ζ92    linear of order 9
ρ1933-1-13333-1-1-1-1000000000000    orthogonal lifted from A4
ρ203-31-133-3-3-1-111000000000000    orthogonal lifted from C2×A4
ρ2133-1-1-3-3√-3/2-3+3√-3/2-3+3√-3/2-3-3√-3/2ζ65ζ6ζ6ζ65000000000000    complex lifted from C3.A4
ρ223-31-1-3+3√-3/2-3-3√-3/23+3√-3/23-3√-3/2ζ6ζ65ζ3ζ32000000000000    complex faithful
ρ233-31-1-3-3√-3/2-3+3√-3/23-3√-3/23+3√-3/2ζ65ζ6ζ32ζ3000000000000    complex faithful
ρ2433-1-1-3+3√-3/2-3-3√-3/2-3-3√-3/2-3+3√-3/2ζ6ζ65ζ65ζ6000000000000    complex lifted from C3.A4

Permutation representations of C2×C3.A4
►On 18 points - transitive group 18T26
Generators in S18
(1 16)(2 17)(3 18)(4 10)(5 11)(6 12)(7 13)(8 14)(9 15)

(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)

(2 17)(3 18)(5 11)(6 12)(8 14)(9 15)

(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)

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

C2×C3.A4 is a maximal subgroup of
 C6.S4  A4×C18  C2.(C42⋊C9)  C24⋊C18  C42⋊C18  C42⋊2C18  C22⋊(Q8⋊C9)  2+ 1+4⋊2C9
C2×C3.A4 is a maximal quotient of
 Q8.C18  C24⋊C18  C42⋊C18  C42⋊2C18

Matrix representation of C2×C3.A4 ►in GL3(𝔽7) generated by

600
060
006
,
400
040
004
,
600
010
006
,
600
060
001
,
004
600
060
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,0,0,0,4,0,0,0,4],[6,0,0,0,1,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,6,0,0,0,6,4,0,0] >;
 

C2×C3.A4 in GAP, Magma, Sage, TeX 

C_2\times C_3.A_4
 
% in TeX
 
G:=Group("C2xC3.A4");
 
// GroupNames label
 
G:=SmallGroup(72,16);
 
// by ID
 
G=gap.SmallGroup(72,16);
 
# by ID
 
G:=PCGroup([5,-2,-3,-3,-2,2,36,368,684]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^2=1,e^3=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
 
// generators/relations
 

Export

Subgroup lattice of C2×C3.A4 in TeX
Character table of C2×C3.A4 in TeX

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