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C2×ΓL(1,𝔽8)×C361 #29503080

order15162
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C361 · Schur multiplier (H2) = C1

Aliases

direct productC2×ΓL(1,𝔽8)×C361
unambiguous semidirect product(C2×C7)⋊(C3×C361)

Splittings

Fitting, HallC2×C7×C361 ⋊ C3
derived, SylowC7 ⋊ C2×C3×C361
HallC7×C361 ⋊ C2×C3
HallC2×C7 ⋊ C3×C361

Subgroups

subgroupG/H
center, hypercenterC2×C361ΓL(1,𝔽8)
commutator, hypocenterC7C2×C3×C361
FrattiniC19C2×C19×ΓL(1,𝔽8)
FittingC2×C7×C361C3
socleC2×C7×C19C3×C19
prime radicalC2×C19×ΓL(1,𝔽8)C19
cosocleC7×C19C2×C3×C19
LCSG ▷ C7
UCSC1 ◁ C2×C361

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5054

Conjugacy classes & character table

3610 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofΓL(1,𝔽8)×Dic361, ΓL(1,𝔽8)⋊C4×C361, ΓL(1,𝔽8)⋊Dic361, …
maximal normal subgroup ofΓL(1,𝔽8)×Dic361, ΓL(1,𝔽8)⋊C4×C361, ΓL(1,𝔽8)⋊Dic361, …
maximal quotient ofC4×ΓL(1,𝔽8)×C361, C22×ΓL(1,𝔽8)×C361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15162, 15);