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C5×ΓL(1,𝔽8)×D37⋊C3 #37552040

order23310
automorphism?
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×ΓL(1,𝔽8)×D37⋊C3
unambiguous semidirect product(C37⋊C3)⋊(C2×C5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D37⋊C3)⋊(C3×C5)
unambiguous semidirect product(ΓL(1,𝔽8)×D37)⋊(C3×C5)
unambiguous semidirect product#13095531⋊(C3×C5)
unambiguous semidirect product#13095532⋊(C3×C5)
unambiguous semidirect product(C5×C7×D37⋊C3)⋊C3
unambiguous semidirect product(C5×ΓL(1,𝔽8)×D37)⋊C3
unambiguous semidirect product(C5×#13095531)⋊C3
unambiguous semidirect product(C5×#13095532)⋊C3

Splittings

Fitting, socle, HallC5×C7×C37 ⋊ C2×C32
derived, HallC7×C37 ⋊ C2×C32×C5
HallC5×ΓL(1,𝔽8)×C37⋊C3 ⋊ C2
HallC5×C7×D37 ⋊ C32
HallΓL(1,𝔽8)×C37⋊C3 ⋊ C2×C5
HallC7×D37 ⋊ C32×C5
HallC5×D37 ⋊ C3×ΓL(1,𝔽8)
HallC5×C37 ⋊ C2×C3×ΓL(1,𝔽8)
HallD37 ⋊ C3×C5×ΓL(1,𝔽8)
SylowC37 ⋊ C2×C3×C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×D37⋊C3
SylowC7 ⋊ C3×C5×D37⋊C3

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)×D37⋊C3
commutator, cosocle, hypocenterC7×C37C2×C32×C5
FrattiniC1G
Fitting, socleC5×C7×C37C2×C32
LCSG ▷ C7×C37
UCSC1 ◁ C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1295

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 59);