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C37×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #45657430

order32634
automorphism?
homologyAbelianization (H1) = C2×C32×C37 · Schur multiplier (H2) = C3

Aliases

direct productC37×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C37×AGL(1,𝔽7))
unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊(C3×C37)
unambiguous semidirect product#64424⋊(C3×C37)
unambiguous semidirect product#64425⋊(C3×C37)

Splittings

Fitting, socle, HallC72×C37 ⋊ C2×C32
derived, SylowC72 ⋊ C2×C32×C37
HallΓL(1,𝔽8)2×C37 ⋊ C2
HallC37×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×C37
HallC7⋊D7 ⋊ C32×C37

Subgroups

subgroupG/H
center, hypercenterC37ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC72C2×C32×C37
FrattiniC1G
Fitting, socleC72×C37C2×C32
LCSG ▷ C72
UCSC1 ◁ C37

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 777

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32634, 210);