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C5×C22⋊((C31⋊C3)⋊C5) #22604841

order9300
automorphism?
homologyAbelianization (H1) = C3×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC5×C22⋊((C31⋊C3)⋊C5)
unambiguous semidirect productC22⋊(C5×(C31⋊C3)⋊C5)
unambiguous semidirect product(C22×C5)⋊((C31⋊C3)⋊C5)
unambiguous semidirect productΓL(1,𝔽32)⋊(C5×A4)
unambiguous semidirect product(C22⋊(C31⋊C3))⋊C52
unambiguous semidirect product(C5×ΓL(1,𝔽32))⋊A4
unambiguous semidirect product(C5×C22⋊(C31⋊C3))⋊C5

Splittings

Fitting, socleC22×C5×C31 ⋊ C3×C5
derived, HallC22×C31 ⋊ C3×C52
HallC22×C5×ΓL(1,𝔽32) ⋊ C3
HallC5×ΓL(1,𝔽32) ⋊ A4
HallC22⋊(C31⋊C3) ⋊ C52
SylowC31 ⋊ C52×A4
SylowC22 ⋊ C5×(C31⋊C3)⋊C5

Subgroups

subgroupG/H
center, hypercenterC5C22⋊((C31⋊C3)⋊C5)
commutator, hypocenterC22×C31C3×C52
FrattiniC1G
Fitting, socleC22×C5×C31C3×C5
LCSG ▷ C22×C31
UCSC1 ◁ C5

80 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×A4×(C31⋊C3)⋊C5, C3×C5×C22⋊((C31⋊C3)⋊C5)
maximal normal subgroup ofC5×A4×(C31⋊C3)⋊C5, C3×C5×C22⋊((C31⋊C3)⋊C5)
maximal quotient ofC5×C22⋊(ΓL(1,𝔽32)⋊C9), C3×C5×C22⋊((C31⋊C3)⋊C5)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9300, 115);