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C22×C3×C5×ΓL(1,𝔽8) #11929565

order1260
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC22×C3×C5×ΓL(1,𝔽8)

Splittings

Fitting, socleC22×C3×C5×C7 ⋊ C3
derived, SylowC7 ⋊ C22×C32×C5
HallC22×C5×C7 ⋊ C32
HallC5×C7 ⋊ C22×C32
HallC22×C7 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC22×C3×C5ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C32×C5
FrattiniC1G
Fitting, socleC22×C3×C5×C7C3
LCSG ▷ C7
UCSC1 ◁ C22×C3×C5

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 420

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×C3×ΓL(1,𝔽8)×AGL(1,𝔽11), C22×C3×ΓL(1,𝔽8)×C11⋊C5, C2×C3×C5×ΓL(1,𝔽8)×D11, …
maximal normal subgroup ofC22×C3×C5×C11×ΓL(1,𝔽8), C22×C3×C5×C13×ΓL(1,𝔽8), C22×C3×C5×C17×ΓL(1,𝔽8), …
maximal quotient ofC2×C3×ΓL(1,𝔽8)×AGL(1,𝔽11), C22×C3×ΓL(1,𝔽8)×C11⋊C5, C2×C3×C5×ΓL(1,𝔽8)×D11, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 117);