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C22×C112×ΓL(1,𝔽8) #23779076

order10164
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C112 · Schur multiplier (H2) = C2×C11

Aliases

direct productC22×C112×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C7×C112 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C112
HallC7×C112 ⋊ C22×C3
HallC22×C7 ⋊ C3×C112

Subgroups

subgroupG/H
center, hypercenterC22×C112ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C112
FrattiniC1G
Fitting, socleC22×C7×C112C3
LCSG ▷ C7
UCSC1 ◁ C22×C112

280 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 ofΓL(1,𝔽8)×C22⋊(C112⋊C3), C112×A4×ΓL(1,𝔽8), C22×ΓL(1,𝔽8)×C112⋊C3, …
maximal normal subgroup ofΓL(1,𝔽8)×C22⋊(C112⋊C3), C112×A4×ΓL(1,𝔽8), C22×ΓL(1,𝔽8)×C112⋊C3, …
maximal quotient ofC22×C112×C7⋊C9, C2×D3×C112×ΓL(1,𝔽8), C22×C3×C112×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10164, 128);