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C4×C112×ΓL(1,𝔽8) #23778998

order10164
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C4×C112 · Schur multiplier (H2) = C11

Aliases

direct productC4×C112×ΓL(1,𝔽8)
unambiguous semidirect product(C4×C7)⋊(C3×C112)
unambiguous semidirect product(C4×C7×C11)⋊(C3×C11)

Splittings

Fitting, HallC4×C7×C112 ⋊ C3
derived, SylowC7 ⋊ C3×C4×C112
HallC7×C112 ⋊ C3×C4
HallC4×C7 ⋊ C3×C112

Subgroups

subgroupG/H
center, hypercenterC4×C112ΓL(1,𝔽8)
commutator, hypocenterC7C3×C4×C112
FrattiniC2C2×C112×ΓL(1,𝔽8)
FittingC4×C7×C112C3
socleC2×C7×C112C2×C3
prime radicalC2×C112×ΓL(1,𝔽8)C2
cosocleC2×C7C2×C3×C112
LCSG ▷ C7
UCSC1 ◁ C4×C112

168 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2420 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC4×ΓL(1,𝔽8)×C112⋊C3, C112×Dic3×ΓL(1,𝔽8), C3×C4×C112×ΓL(1,𝔽8)
maximal normal subgroup ofC4×ΓL(1,𝔽8)×C112⋊C3, C3×C4×C112×ΓL(1,𝔽8)
maximal quotient ofC4×C112×C7⋊C9, C112×Dic3×ΓL(1,𝔽8), C3×C4×C112×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10164, 50);