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C5×ΓL(1,𝔽8)×C61⋊C5 #44545059

order32025
automorphism?
homologyAbelianization (H1) = C3×C52 · Schur multiplier (H2) = C5

Aliases

direct productC5×ΓL(1,𝔽8)×C61⋊C5
unambiguous semidirect productC7⋊(C3×C5×C61⋊C5)
unambiguous semidirect product(C5×C7)⋊(C3×C61⋊C5)
unambiguous semidirect product(ΓL(1,𝔽8)×C61)⋊C52
unambiguous semidirect product(C5×ΓL(1,𝔽8)×C61)⋊C5

Splittings

Fitting, socleC5×C7×C61 ⋊ C3×C5
derived, HallC7×C61 ⋊ C3×C52
HallC5×C7×C61⋊C5 ⋊ C3
HallΓL(1,𝔽8)×C61 ⋊ C52
SylowC61 ⋊ C52×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C5×C61⋊C5

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)×C61⋊C5
commutator, cosocle, hypocenterC7×C61C3×C52
FrattiniC1G
Fitting, socleC5×C7×C61C3×C5
LCSG ▷ C7×C61
UCSC1 ◁ C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2135

Conjugacy classes & character table

425 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32025, 15);