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Cayley diagram

A4×C25 #64465

order300
isoclinismisoclinic class of A4
automorphismAut(G) = C4×C5×S4 (order 480) · Inn(G) = A4 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C3×C25 · Schur multiplier (H2) = C2

Aliases

direct productA4×C25
unambiguous semidirect productC22⋊(C3×C25)
unambiguous semidirect product(C22×C25)⋊C3

Splittings

Fitting, HallC22×C25 ⋊ C3
derived, SylowC22 ⋊ C3×C25

Subgroups

subgroupG/H
center, hypercenterC25A4
commutator, hypocenterC22C3×C25
FrattiniC5C5×A4
FittingC22×C25C3
socle, cosocleC22×C5C3×C5
prime radicalC5×A4C5
LCSG ▷ C22
UCSC1 ◁ C25

15 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

(2,3,4)(5,6,8,11,15,7,9,12,16,20,10,13,17,21,24,14,18,22,25,27,19,23,26,28,29);(1,4)(2,3)

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup ofA4⋊D25, A4⋊Dic25, A4⋊#1035, …
maximal subgroup ofS4×C25, A4⋊D25, A4×D25, …
maximal normal subgroup ofS4×C25, A4⋊D25, A4×D25, …
maximal quotient ofSL(2,𝔽3)×C25, C2×A4×C25, C25×C22⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 8);