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C2×C3×C5×ΓL(1,𝔽8)×C37 #37552124

order23310
automorphism?
homologyAbelianization (H1) = C2×C32×C5×C37 · Schur multiplier (H2) = C3

Aliases

direct productC2×C3×C5×ΓL(1,𝔽8)×C37
unambiguous semidirect product(C2×C7)⋊(C32×C5×C37)
unambiguous semidirect product(C2×C3×C7)⋊(C3×C5×C37)
unambiguous semidirect product(C2×C5×C7)⋊(C32×C37)
unambiguous semidirect product(C2×C3×C5×C7)⋊(C3×C37)

Splittings

Fitting, socleC2×C3×C5×C7×C37 ⋊ C3
derived, SylowC7 ⋊ C2×C32×C5×C37
HallC2×C5×C7×C37 ⋊ C32
HallC5×C7×C37 ⋊ C2×C32
HallC2×C7×C37 ⋊ C32×C5
HallC7×C37 ⋊ C2×C32×C5
HallC2×C5×C7 ⋊ C32×C37
HallC5×C7 ⋊ C2×C32×C37
HallC2×C7 ⋊ C32×C5×C37

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5×C37ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C32×C5×C37
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C37C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5×C37

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7770

Conjugacy classes & character table

5550 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 143);