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C22×C7×C17×ΓL(1,𝔽8) #23756270

order9996
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C7×C17 · Schur multiplier (H2) = C2

Aliases

direct productC22×C7×C17×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C72×C17 ⋊ C3
derivedC7 ⋊ C22×C3×C7×C17
HallC72×C17 ⋊ C22×C3
HallC22×C72 ⋊ C3×C17
SylowC72 ⋊ C22×C3×C17

Subgroups

subgroupG/H
center, hypercenterC22×C7×C17ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C7×C17
FrattiniC1G
Fitting, socleC22×C72×C17C3
LCSG ▷ C7
UCSC1 ◁ C22×C7×C17

100 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

2380 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×A4×C17×ΓL(1,𝔽8), C17×ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8), C22×C17×ΓL(1,𝔽8)2, …
maximal normal subgroup ofC7×A4×C17×ΓL(1,𝔽8), C17×ΓL(1,𝔽8)×C22⋊ΓL(1,𝔽8), C22×C17×ΓL(1,𝔽8)2, …
maximal quotient ofC22×C7×C17×C7⋊C9, C2×D3×C7×C17×ΓL(1,𝔽8), C22×C3×C7×C17×ΓL(1,𝔽8)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 173);