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C5×C3⋊SL(2,ℤ/4) #10624065

order720
isoclinismisoclinic class of C3⋊S4
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×C3⋊SL(2,ℤ/4)
unambiguous semidirect productC3⋊(C5×SL(2,ℤ/4))
unambiguous semidirect productA4⋊(C5×Dic3)

Splittings

derivedC3×A4 ⋊ C4×C5

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊S4
commutator, hypocenterC3×A4C4×C5
FrattiniC2C5×C3⋊S4
Fitting, socleC23×C3×C5D3
prime radicalC2×C3×C5×A4C2
cosocleC2×C3×A4C2×C5
LCSG ▷ C3×A4
UCSC1 ◁ C2×C5

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 180

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,3,4,5)(6,7,8,9)(10,11,13,12)(15,16);(14,15,16);(11,12,13)

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×#64204, C5×Dic3×S4, C5×D3×SL(2,ℤ/4), …
maximal normal subgroup ofC5×#64204, C5×Dic3×S4, C5×D3×SL(2,ℤ/4), …
maximal quotient ofC5×C3⋊(A4⋊C8), C5×C3⋊#859, C5×C3⋊GU(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(720, 534);