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C3×C5×C37×AGL(1,𝔽7) #37552064

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

Aliases

direct productC3×C5×C37×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C32×C5×C37)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C3×C5×C37)
unambiguous semidirect product(C3×D7)⋊(C3×C5×C37)
unambiguous semidirect product(C5×D7)⋊(C32×C37)
unambiguous semidirect product(C3×C5×D7)⋊(C3×C37)

Splittings

Fitting, socleC3×C5×C7×C37 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C32×C5×C37
HallC3×C5×ΓL(1,𝔽8)×C37 ⋊ C2
HallC5×D7×C37 ⋊ C32
HallC3×ΓL(1,𝔽8)×C37 ⋊ C2×C5
HallC5×C7×C37 ⋊ C2×C32
HallD7×C37 ⋊ C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×C37
HallC7×C37 ⋊ C2×C32×C5
HallC5×D7 ⋊ C32×C37
HallC3×ΓL(1,𝔽8) ⋊ C2×C5×C37
HallC5×C7 ⋊ C2×C32×C37
HallD7 ⋊ C32×C5×C37

Subgroups

subgroupG/H
center, hypercenterC3×C5×C37AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C2×C32×C5×C37
FrattiniC1G
Fitting, socleC3×C5×C7×C37C2×C3
LCSG ▷ C7
UCSC1 ◁ C3×C5×C37

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3885

Conjugacy classes & character table

3885 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23310, 83);