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C5×ΓL(1,𝔽8)×C167 #31776279

order17535
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C5×C167 · Schur multiplier (H2) = C1

Aliases

direct productC5×ΓL(1,𝔽8)×C167
unambiguous semidirect productC7⋊(C3×C5×C167)
unambiguous semidirect product(C5×C7)⋊(C3×C167)
unambiguous semidirect product(C7×C167)⋊(C3×C5)
unambiguous semidirect product(C5×C7×C167)⋊C3

Splittings

Fitting, socle, HallC5×C7×C167 ⋊ C3
derived, SylowC7 ⋊ C3×C5×C167
HallC7×C167 ⋊ C3×C5
HallC5×C7 ⋊ C3×C167

Subgroups

subgroupG/H
center, hypercenterC5×C167ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C3×C5×C167
FrattiniC1G
Fitting, socleC5×C7×C167C3
LCSG ▷ C7
UCSC1 ◁ C5×C167

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5845

Conjugacy classes & character table

4175 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C5×ΓL(1,𝔽8)×C167, C5×ΓL(1,𝔽8)×D167, C5×AGL(1,𝔽7)×C167, …
maximal normal subgroup ofC2×C5×ΓL(1,𝔽8)×C167, C5×ΓL(1,𝔽8)×D167, C5×AGL(1,𝔽7)×C167, …
maximal quotient ofC2×C5×ΓL(1,𝔽8)×C167

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17535, 2);