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ΓL(1,𝔽8)×C25×C31 #30653673

order16275
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C25×C31 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×C25×C31
unambiguous semidirect productC7⋊(C3×C25×C31)
unambiguous semidirect product(C7×C25)⋊(C3×C31)

Splittings

Fitting, HallC7×C25×C31 ⋊ C3
derived, SylowC7 ⋊ C3×C25×C31
HallC7×C31 ⋊ C3×C25
HallC7×C25 ⋊ C3×C31

Subgroups

subgroupG/H
center, hypercenterC25×C31ΓL(1,𝔽8)
commutator, hypocenterC7C3×C25×C31
FrattiniC5C5×ΓL(1,𝔽8)×C31
FittingC7×C25×C31C3
socleC5×C7×C31C3×C5
prime radicalC5×ΓL(1,𝔽8)×C31C5
cosocleC5×C7C3×C5×C31
LCSG ▷ C7
UCSC1 ◁ C25×C31

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5425

Conjugacy classes & character table

3875 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×ΓL(1,𝔽8)×C25×C31, ΓL(1,𝔽8)×C25×D31, C25×C31×AGL(1,𝔽7), …
maximal normal subgroup ofC2×ΓL(1,𝔽8)×C25×C31, ΓL(1,𝔽8)×C25×D31, C25×C31×AGL(1,𝔽7), …
maximal quotient ofC2×ΓL(1,𝔽8)×C25×C31

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16275, 4);