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C22×C11×C13×ΓL(1,𝔽8) #26046582

order12012
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C11×C13 · Schur multiplier (H2) = C2

Aliases

direct productC22×C11×C13×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C7×C11×C13 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C11×C13
HallC7×C11×C13 ⋊ C22×C3
HallC22×C7×C13 ⋊ C3×C11
HallC22×C7×C11 ⋊ C3×C13
HallC7×C13 ⋊ C22×C3×C11
HallC7×C11 ⋊ C22×C3×C13
HallC22×C7 ⋊ C3×C11×C13

Subgroups

subgroupG/H
center, hypercenterC22×C11×C13ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C11×C13
FrattiniC1G
Fitting, socleC22×C7×C11×C13C3
LCSG ▷ C7
UCSC1 ◁ C22×C11×C13

80 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2860 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC11×A4×C13×ΓL(1,𝔽8), C11×ΓL(1,𝔽8)×C22⋊(C13⋊C3), C22×C11×ΓL(1,𝔽8)×C13⋊C3, …
maximal normal subgroup ofC11×A4×C13×ΓL(1,𝔽8), C11×ΓL(1,𝔽8)×C22⋊(C13⋊C3), C22×C11×ΓL(1,𝔽8)×C13⋊C3, …
maximal quotient ofC22×C11×C13×C7⋊C9, C2×D3×C11×C13×ΓL(1,𝔽8), C22×C3×C11×C13×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12012, 172);