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Cayley diagram

A4×3−1+2 #66439

order324
automorphism?
homologyAbelianization (H1) = C33 · Schur multiplier (H2) = C2×C32

Aliases

direct productA4×3−1+2

Splittings

FittingC22×3−1+2 ⋊ C3
SylowC22 ⋊ C3×3−1+2

Subgroups

subgroupG/H
center, FrattiniC3C32×A4
commutator, socle, cosocleC22×C3C33
FittingC22×3−1+2C3
prime radicalC32×A4C3
hypercenter3−1+2A4
hypocenterC22C3×3−1+2
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C3 ◁ 3−1+2

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

(1,4)(2,3)(6,9,10)(8,12,13);(5,6,8,7,10,12,11,9,13);(2,3,4)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×3−1+2, A4⋊AGL(1,ℤ/9), A4×AGL(1,ℤ/9), …
maximal normal subgroup ofS4×3−1+2, A4⋊AGL(1,ℤ/9), A4×AGL(1,ℤ/9), …
maximal quotient ofSL(2,𝔽3)×3−1+2, C2×A4×3−1+2, #11757767, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 131);