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C4×C11×C13×ΓL(1,𝔽8) #26046413

order12012
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C11×C13 · Schur multiplier (H2) = C1

Aliases

direct productC4×C11×C13×ΓL(1,𝔽8)
unambiguous semidirect product(C4×C7)⋊(C3×C11×C13)
unambiguous semidirect product(C4×C7×C11)⋊(C3×C13)

Splittings

Fitting, HallC4×C7×C11×C13 ⋊ C3
derived, SylowC7 ⋊ C3×C4×C11×C13
HallC7×C11×C13 ⋊ C3×C4
HallC4×C7×C13 ⋊ C3×C11
HallC4×C7×C11 ⋊ C3×C13
HallC7×C13 ⋊ C3×C4×C11
HallC7×C11 ⋊ C3×C4×C13
HallC4×C7 ⋊ C3×C11×C13

Subgroups

subgroupG/H
center, hypercenterC4×C11×C13ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C11×C13
FrattiniC2C2×C11×C13×ΓL(1,𝔽8)
FittingC4×C7×C11×C13C3
socleC2×C7×C11×C13C2×C3
prime radicalC2×C11×C13×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C11×C13
LCSG ▷ C7
UCSC1 ◁ C4×C11×C13

48 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 ofC4×C11×ΓL(1,𝔽8)×C13⋊C3, C11×Dic3×C13×ΓL(1,𝔽8), C3×C4×C11×C13×ΓL(1,𝔽8)
maximal normal subgroup ofC4×C11×ΓL(1,𝔽8)×C13⋊C3, C3×C4×C11×C13×ΓL(1,𝔽8)
maximal quotient ofC4×C11×C13×C7⋊C9, C11×Dic3×C13×ΓL(1,𝔽8), C3×C4×C11×C13×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12012, 3);