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G = C32⋊A4order 108 = 22·33

The semidirect product of C32 and A4 acting via A4/C22=C3

metabelian, soluble, monomial

Aliases: C32⋊A4, C22⋊He3, C621C3, (C3×A4)⋊C3, C3.5(C3×A4), (C2×C6).5C32, SmallGroup(108,22)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C32⋊A4
C1C22C2×C6C3×A4 — C32⋊A4
C22C2×C6 — C32⋊A4
C1C3C32

Generators and relations for C32⋊A4
 G = < a,b,c,d,e | a3=b3=c2=d2=e3=1, ab=ba, ac=ca, ad=da, eae-1=ab-1, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >

3C2
3C3
12C3
12C3
12C3
3C6
3C6
3C6
3C6
4C32
4C32
4C32
3A4
3A4
3C2×C6
3A4
3C3×C6
4He3

Character table of C32⋊A4

 class 123A3B3C3D3E3F3G3H3I3J6A6B6C6D6E6F6G6H
 size 13113312121212121233333333
ρ111111111111111111111    trivial
ρ2111111ζ3ζ32ζ32ζ32ζ3ζ311111111    linear of order 3
ρ31111ζ3ζ321ζ31ζ32ζ32ζ3ζ31ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ41111ζ32ζ3ζ3ζ3ζ321ζ321ζ321ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ51111ζ32ζ3ζ321ζ3ζ321ζ3ζ321ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ61111ζ3ζ32ζ32ζ32ζ31ζ31ζ31ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ71111ζ32ζ31ζ321ζ3ζ3ζ32ζ321ζ3ζ3ζ31ζ32ζ32    linear of order 3
ρ8111111ζ32ζ3ζ3ζ3ζ32ζ3211111111    linear of order 3
ρ91111ζ3ζ32ζ31ζ32ζ31ζ32ζ31ζ32ζ32ζ321ζ3ζ3    linear of order 3
ρ103-13333000000-1-1-1-1-1-1-1-1    orthogonal lifted from A4
ρ113-1-3-3-3/2-3+3-3/200000000-1--3ζ6-1--32-1+-3ζ65-1+-32    complex faithful
ρ123-1-3-3-3/2-3+3-3/2000000002ζ6-1+-3-1--32ζ65-1--3-1+-3    complex faithful
ρ1333-3+3-3/2-3-3-3/2000000000-3+3-3/2000-3-3-3/200    complex lifted from He3
ρ143-1-3+3-3/2-3-3-3/2000000002ζ65-1--3-1+-32ζ6-1+-3-1--3    complex faithful
ρ153-1-3+3-3/2-3-3-3/200000000-1+-3ζ65-1+-32-1--3ζ6-1--32    complex faithful
ρ163-133-3-3-3/2-3+3-3/2000000ζ6-1ζ65ζ65ζ65-1ζ6ζ6    complex lifted from C3×A4
ρ1733-3-3-3/2-3+3-3/2000000000-3-3-3/2000-3+3-3/200    complex lifted from He3
ρ183-1-3-3-3/2-3+3-3/200000000-1+-3ζ62-1+-3-1--3ζ652-1--3    complex faithful
ρ193-1-3+3-3/2-3-3-3/200000000-1--3ζ652-1--3-1+-3ζ62-1+-3    complex faithful
ρ203-133-3+3-3/2-3-3-3/2000000ζ65-1ζ6ζ6ζ6-1ζ65ζ65    complex lifted from C3×A4

Permutation representations of C32⋊A4
On 18 points - transitive group 18T48
Generators in S18
(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 5 3)(2 6 4)(7 9 8)(10 11 12)(13 14 15)(16 18 17)
(1 2)(3 4)(5 6)(10 15)(11 13)(12 14)
(1 2)(3 4)(5 6)(7 18)(8 16)(9 17)
(1 17 13)(2 9 11)(3 18 15)(4 7 10)(5 16 14)(6 8 12)

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

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

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

G:=TransitiveGroup(18,48);

C32⋊A4 is a maximal subgroup of
C62⋊S3  C32⋊S4  C62⋊C6  C62.13C32  C62.14C32  He3⋊A4  3- 1+2⋊A4  C62.6C32  C332A4  C62.25C32  A4×He3  C62.9C32  C42⋊He3  C24⋊He3  C62⋊A4
C32⋊A4 is a maximal quotient of
Q8⋊He3  C62.13C32  C62.14C32  C62.15C32  C62.16C32  He3.A4  He3⋊A4  He32A4  C62.C32  3- 1+2⋊A4  C62.6C32  C62⋊C9  C332A4  C42⋊He3  C24⋊He3  C62⋊A4

Matrix representation of C32⋊A4 in GL3(𝔽7) generated by

100
020
004
,
200
020
002
,
600
010
006
,
100
060
006
,
010
001
100
G:=sub<GL(3,GF(7))| [1,0,0,0,2,0,0,0,4],[2,0,0,0,2,0,0,0,2],[6,0,0,0,1,0,0,0,6],[1,0,0,0,6,0,0,0,6],[0,0,1,1,0,0,0,1,0] >;

C32⋊A4 in GAP, Magma, Sage, TeX

C_3^2\rtimes A_4
% in TeX

G:=Group("C3^2:A4");
// GroupNames label

G:=SmallGroup(108,22);
// by ID

G=gap.SmallGroup(108,22);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,121,1083,2029]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^-1,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

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Subgroup lattice of C32⋊A4 in TeX
Character table of C32⋊A4 in TeX

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