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Cayley diagram

A4×AGL(1,𝔽5) #6832

order240
isoclinismisoclinic class of A4×AGL(1,𝔽5)
automorphismAut(G) = AGL(1,𝔽5)×S4 (order 480) · Inn(G) = G · Out(G) = C2
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productA4×AGL(1,𝔽5)
unambiguous semidirect product(C22×AGL(1,𝔽5))⋊C3

Splittings

Fitting, socle, derivedC22×C5 ⋊ C3×C4
HallC22×AGL(1,𝔽5) ⋊ C3
SylowC5 ⋊ C4×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC22×C5C3×C4
prime radicalD5×A4C2
cosocleC22×D5C2×C3
LCSG ▷ C22×C5
UCSC1

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 9 points:

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

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×S4, C2×A4×AGL(1,𝔽5), C3×A4×AGL(1,𝔽5), …
maximal normal subgroup ofAGL(1,𝔽5)×S4, C2×A4×AGL(1,𝔽5), C3×A4×AGL(1,𝔽5), …
maximal quotient of#4133⋊C3, AGL(1,𝔽5)×SL(2,𝔽3), A4×#170, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 193);