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Cayley diagram

A4×H(3,𝔽3) #66438

order324
automorphism?
homologyAbelianization (H1) = C33 · Schur multiplier (H2) = C2×C34

Aliases

direct productA4×H(3,𝔽3)

Splittings

FittingC22×H(3,𝔽3) ⋊ C3
SylowC22 ⋊ C3×H(3,𝔽3)

Subgroups

subgroupG/H
center, FrattiniC3C32×A4
commutator, socle, cosocleC22×C3C33
FittingC22×H(3,𝔽3)C3
hypercenterH(3,𝔽3)A4
hypocenterC22C3×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C3 ◁ H(3,𝔽3)

123 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)(9,13,11);(2,3,4)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4⋊#272, A4⋊#1084, A4⋊#1088, …
maximal subgroup ofS4×H(3,𝔽3), A4⋊((C3⋊D3)⋊C3), A4⋊#272, …
maximal normal subgroup ofS4×H(3,𝔽3), A4⋊((C3⋊D3)⋊C3), A4⋊#272, …
maximal quotient ofSL(2,𝔽3)×H(3,𝔽3), C2×A4×H(3,𝔽3), A4×C32⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(324, 130);