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S4×C25 #10597053

order600
isoclinismisoclinic class of S4
automorphism?
homologyAbelianization (H1) = C2×C25 · Schur multiplier (H2) = C2

Aliases

direct productS4×C25
unambiguous semidirect productA4⋊(C2×C25)

Splittings

FittingC22×C25 ⋊ D3
derivedA4 ⋊ C2×C25

Subgroups

subgroupG/H
center, hypercenterC25S4
commutator, hypocenterA4C2×C25
FrattiniC5C5×S4
FittingC22×C25D3
socleC22×C5C5×D3
prime radicalC5×S4C5
cosocleC5×A4C2×C5
LCSG ▷ A4
UCSC1 ◁ C25

33 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:

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

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofS4×D25, C2×S4×C25, C3×S4×C25, …
maximal normal subgroup ofS4×D25, C2×S4×C25, C3×S4×C25, …
maximal quotient ofC25×#226, C25×GL(2,𝔽3), C25×SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 34);