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C4×C11×ΓL(1,𝔽8) #11745103

order924
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C11 · Schur multiplier (H2) = C1

Aliases

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

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC4×C11ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C11
FrattiniC2C2×C11×ΓL(1,𝔽8)
FittingC4×C7×C11C3
socleC2×C7×C11C2×C3
prime radicalC2×C11×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C11
LCSG ▷ C7
UCSC1 ◁ C4×C11

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 308

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×C11⋊C8, C11×ΓL(1,𝔽8)⋊C8, C11⋊(ΓL(1,𝔽8)⋊C8), …
maximal normal subgroup ofΓL(1,𝔽8)×C11⋊C8, C11×ΓL(1,𝔽8)⋊C8, C11⋊(ΓL(1,𝔽8)⋊C8), …
maximal quotient ofC8×C11×ΓL(1,𝔽8), C2×C4×C11×ΓL(1,𝔽8), C4×C11×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(924, 4);