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G = C3.A4  order 36 = 22·32

The central extension by C3 of A4

metabelian, soluble, monomial, A-group

Aliases: C3.A4, C22⋊C9, (C2×C6).C3, SmallGroup(36,3)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3.A4
 G = < a,b,c,d | a3=b2=c2=1, d3=a, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, dcd-1=b >

C1
3C2
C3
C22
3C6
4C9
C2×C6
C3.A4

Character table of C3.A4

 class 123A3B6A6B9A9B9C9D9E9F
 size 131133444444
ρ1111111111111    trivial
ρ2111111ζ32ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ3111111ζ3ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ411ζ32ζ3ζ3ζ32ζ9ζ95ζ92ζ97ζ94ζ98    linear of order 9
ρ511ζ32ζ3ζ3ζ32ζ97ζ98ζ95ζ94ζ9ζ92    linear of order 9
ρ611ζ3ζ32ζ32ζ3ζ92ζ9ζ94ζ95ζ98ζ97    linear of order 9
ρ711ζ3ζ32ζ32ζ3ζ98ζ94ζ97ζ92ζ95ζ9    linear of order 9
ρ811ζ32ζ3ζ3ζ32ζ94ζ92ζ98ζ9ζ97ζ95    linear of order 9
ρ911ζ3ζ32ζ32ζ3ζ95ζ97ζ9ζ98ζ92ζ94    linear of order 9
ρ103-133-1-1000000    orthogonal lifted from A4
ρ113-1-3+3√-3/2-3-3√-3/2ζ6ζ65000000    complex faithful
ρ123-1-3-3√-3/2-3+3√-3/2ζ65ζ6000000    complex faithful

Permutation representations of C3.A4
►On 18 points - transitive group 18T7
Generators in S18
(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)| (1,4,7)(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( (1,4,7)(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([[(1,4,7),(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(18,7);
 

C3.A4 is a maximal subgroup of
 C3.S4  C9×A4  C9⋊A4  C32.A4  C42⋊C9  C24⋊C9  C21.A4  C39.A4
C3.A4 is a maximal quotient of
 Q8⋊C9  C9.A4  C42⋊C9  C24⋊C9  C21.A4  C39.A4

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

400
040
004
,
600
010
006
,
600
060
001
,
004
100
010
G:=sub<GL(3,GF(7))| [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],[0,1,0,0,0,1,4,0,0] >;
 

C3.A4 in GAP, Magma, Sage, TeX 

C_3.A_4
 
% in TeX
 
G:=Group("C3.A4");
 
// GroupNames label
 
G:=SmallGroup(36,3);
 
// by ID
 
G=gap.SmallGroup(36,3);
 
# by ID
 
G:=PCGroup([4,-3,-3,-2,2,12,218,435]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^2=1,d^3=a,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 C3.A4 in TeX
Character table of C3.A4 in TeX

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