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Cayley diagram

A5⋊C4 #6730

order240
isoclinismisoclinic class of S5
automorphismAut(G) = C2×S5 (order 240) · Inn(G) = S5 · Out(G) = C2
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productA5⋊C4

Splittings

derivedA5 ⋊ C4

Subgroups

subgroupG/H
center, Frattini, Fitting, hypercenterC2S5
commutator, hypocenterA5C4
socle, prime radical, cosocleC2×A5C2
LCSG ▷ A5
UCSC1 ◁ C2

35 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 9
transitive degreeμt(G) = 12

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

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

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4×S5, A5⋊Q8, C22⋊S5, …
maximal normal subgroup ofC4×S5, A5⋊Q8, C22⋊S5, …
maximal quotient ofA5⋊C8, GL(2,𝔽5), #91143, …
outer automorphism group ofC5⋊D5, #6953

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(240, 91);