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C17×ΓL(1,𝔽8)×C23 #22551367

order8211
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C3×C17×C23 · Schur multiplier (H2) = C1

Aliases

direct productC17×ΓL(1,𝔽8)×C23
unambiguous semidirect productC7⋊(C3×C17×C23)
unambiguous semidirect product(C7×C17)⋊(C3×C23)
unambiguous semidirect product(C7×C23)⋊(C3×C17)
unambiguous semidirect product(C7×C17×C23)⋊C3

Splittings

Fitting, socle, HallC7×C17×C23 ⋊ C3
derived, SylowC7 ⋊ C3×C17×C23
HallC7×C23 ⋊ C3×C17
HallC7×C17 ⋊ C3×C23

Subgroups

subgroupG/H
center, hypercenterC17×C23ΓL(1,𝔽8)
commutator, hypocenterC7C3×C17×C23
FrattiniC1G
Fitting, socleC7×C17×C23C3
LCSG ▷ C7
UCSC1 ◁ C17×C23

16 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1955 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C17×ΓL(1,𝔽8)×C23, C17×ΓL(1,𝔽8)×D23, ΓL(1,𝔽8)×C23×D17, …
maximal normal subgroup ofC2×C17×ΓL(1,𝔽8)×C23, C17×ΓL(1,𝔽8)×D23, ΓL(1,𝔽8)×C23×D17, …
maximal quotient ofC2×C17×ΓL(1,𝔽8)×C23, C17×C23×C7⋊C9, C3×C17×ΓL(1,𝔽8)×C23, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8211, 2);