Copied to
clipboard

G = C32.A4order 108 = 22·33

The non-split extension by C32 of A4 acting via A4/C22=C3

metabelian, soluble, monomial

Aliases: C32.A4, C62.2C3, C2223- 1+2, C3.A42C3, C3.4(C3×A4), (C2×C6).4C32, SmallGroup(108,21)

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=1, e3=b, 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
3C6
3C6
3C6
3C6
4C9
4C9
4C9
3C2×C6
3C3×C6
43- 1+2

Character table of C32.A4

 class 123A3B3C3D6A6B6C6D6E6F6G6H9A9B9C9D9E9F
 size 13113333333333121212121212
ρ111111111111111111111    trivial
ρ21111ζ32ζ31ζ3ζ3ζ31ζ32ζ32ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ311111111111111ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ411111111111111ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ51111ζ3ζ321ζ32ζ32ζ321ζ3ζ3ζ3ζ31ζ32ζ32ζ31    linear of order 3
ρ61111ζ32ζ31ζ3ζ3ζ31ζ32ζ32ζ321ζ3ζ321ζ3ζ32    linear of order 3
ρ71111ζ3ζ321ζ32ζ32ζ321ζ3ζ3ζ31ζ32ζ31ζ32ζ3    linear of order 3
ρ81111ζ3ζ321ζ32ζ32ζ321ζ3ζ3ζ3ζ32ζ31ζ31ζ32    linear of order 3
ρ91111ζ32ζ31ζ3ζ3ζ31ζ32ζ32ζ32ζ321ζ3ζ3ζ321    linear of order 3
ρ103-13333-1-1-1-1-1-1-1-1000000    orthogonal lifted from A4
ρ1133-3-3-3/2-3+3-3/200-3-3-3/2000-3+3-3/2000000000    complex lifted from 3- 1+2
ρ123-1-3-3-3/2-3+3-3/200ζ62-1+-3-1--3ζ652-1--3-1+-3000000    complex faithful
ρ133-1-3-3-3/2-3+3-3/200ζ6-1+-3-1--32ζ65-1--3-1+-32000000    complex faithful
ρ143-133-3-3-3/2-3+3-3/2-1ζ65ζ65ζ65-1ζ6ζ6ζ6000000    complex lifted from C3×A4
ρ153-1-3-3-3/2-3+3-3/200ζ6-1--32-1+-3ζ65-1+-32-1--3000000    complex faithful
ρ163-1-3+3-3/2-3-3-3/200ζ652-1--3-1+-3ζ62-1+-3-1--3000000    complex faithful
ρ1733-3+3-3/2-3-3-3/200-3+3-3/2000-3-3-3/2000000000    complex lifted from 3- 1+2
ρ183-1-3+3-3/2-3-3-3/200ζ65-1+-32-1--3ζ6-1--32-1+-3000000    complex faithful
ρ193-1-3+3-3/2-3-3-3/200ζ65-1--3-1+-32ζ6-1+-3-1--32000000    complex faithful
ρ203-133-3+3-3/2-3-3-3/2-1ζ6ζ6ζ6-1ζ65ζ65ζ65000000    complex lifted from C3×A4

Permutation representations of C32.A4
On 18 points - transitive group 18T47
Generators in S18
(2 8 5)(3 6 9)(11 17 14)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)
(1 16)(3 18)(4 10)(6 12)(7 13)(9 15)
(1 16)(2 17)(4 10)(5 11)(7 13)(8 14)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

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

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

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

G:=TransitiveGroup(18,47);

C32.A4 is a maximal subgroup of
C32.S4  C62.13C32  C62.15C32  He3.A4  He32A4  C62.C32  3- 1+2⋊A4  C62.6C32  C332A4  C62.25C32  He3.2A4  A4×3- 1+2  C62.9C32  C122.C3  C24⋊3- 1+2  C62.A4
C32.A4 is a maximal quotient of
Q8⋊3- 1+2  C62.11C32  C62⋊C9  C122.C3  C24⋊3- 1+2  C62.A4

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

400
020
001
,
200
020
002
,
600
010
006
,
600
060
001
,
004
400
010
G:=sub<GL(3,GF(7))| [4,0,0,0,2,0,0,0,1],[2,0,0,0,2,0,0,0,2],[6,0,0,0,1,0,0,0,6],[6,0,0,0,6,0,0,0,1],[0,4,0,0,0,1,4,0,0] >;

C32.A4 in GAP, Magma, Sage, TeX

C_3^2.A_4
% in TeX

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

G:=SmallGroup(108,21);
// by ID

G=gap.SmallGroup(108,21);
# by ID

G:=PCGroup([5,-3,-3,-3,-2,2,45,121,1083,2029]);
// Polycyclic

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

Export

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

׿
×
𝔽