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C2×C5×ΓL(1,𝔽8)×C37⋊C3 #37552062

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

Aliases

direct productC2×C5×ΓL(1,𝔽8)×C37⋊C3
unambiguous semidirect product(C2×C7×C37⋊C3)⋊(C3×C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8)×C37)⋊(C3×C5)
unambiguous semidirect product(C2×#11715913)⋊(C3×C5)
unambiguous semidirect product(C2×#11715912)⋊(C3×C5)
unambiguous semidirect product(C2×C5×C7×C37⋊C3)⋊C3
unambiguous semidirect product(C2×C5×ΓL(1,𝔽8)×C37)⋊C3
unambiguous semidirect product(C2×C5×#11715913)⋊C3
unambiguous semidirect product(C2×C5×#11715912)⋊C3

Splittings

Fitting, socle, HallC2×C5×C7×C37 ⋊ C32
derived, HallC7×C37 ⋊ C2×C32×C5
HallC5×C7×C37 ⋊ C2×C32
HallC2×C7×C37 ⋊ C32×C5
HallC2×C5×C37 ⋊ C3×ΓL(1,𝔽8)
HallC5×C37 ⋊ C2×C3×ΓL(1,𝔽8)
HallC2×C37 ⋊ C3×C5×ΓL(1,𝔽8)
HallC2×C5×C7 ⋊ C3×C37⋊C3
SylowC37 ⋊ C2×C3×C5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C2×C3×C37⋊C3
HallC2×C7 ⋊ C3×C5×C37⋊C3
SylowC7 ⋊ C2×C3×C5×C37⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)×C37⋊C3
commutator, cosocle, hypocenterC7×C37C2×C32×C5
FrattiniC1G
Fitting, socleC2×C5×C7×C37C32
LCSG ▷ C7×C37
UCSC1 ◁ C2×C5

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2590

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 81);