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Cayley diagram

A4×Dic5 #6749

order240
isoclinismisoclinic class of D5×A4
automorphismAut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = D5×A4 · Out(G) = C23
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C2

Aliases

direct productA4×Dic5
unambiguous semidirect product(C22×Dic5)⋊C3

Splittings

derivedC22×C5 ⋊ C3×C4
HallC22×Dic5 ⋊ C3
SylowC5 ⋊ C4×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D5×A4
commutator, hypocenterC22×C5C3×C4
Fitting, socle, cosocleC23×C5C2×C3
prime radicalC2×C5×A4C2
LCSG ▷ C22×C5
UCSC1 ◁ C2

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 60

A minimal faithful permutation representation — 2 generators on 13 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4⋊#170, A4×#170, #91899, …
maximal normal subgroup ofA4⋊#170, A4×#170, #91899, …
maximal quotient ofA4×#168, Dic5×SL(2,𝔽3), #4094⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 110);