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Cayley diagram

A4×#157 #89097

order432
isoclinismisoclinic class of A4×#157
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productA4×#157
unambiguous semidirect product(C22×#157)⋊C3

Splittings

Fitting, socle, derivedC22×C32 ⋊ C3×C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC22×C32C3×C4
prime radicalA4×C3⋊D3C2
cosocleC22×C3⋊D3C2×C3
LCSG ▷ C22×C32
UCSC1

70 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 10
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 10 points:

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

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

derived subgroup ofAGL(1,𝔽9)⋊S4, #13181779, #13181780, …
maximal subgroup ofA4⋊AGL(1,𝔽9), A4×AGL(1,𝔽9), A4⋊PSU(3,𝔽2), …
maximal normal subgroup ofA4⋊AGL(1,𝔽9), A4×AGL(1,𝔽9), A4⋊PSU(3,𝔽2), …
maximal quotient of#64289⋊C3, SL(2,𝔽3)×#157, A4×#622, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 744);