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C2×C17×ΓL(1,𝔽8)×C23 #30655977

order16422
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C17×C23 · Schur multiplier (H2) = C1

Aliases

direct productC2×C17×ΓL(1,𝔽8)×C23
unambiguous semidirect product(C2×C7)⋊(C3×C17×C23)
unambiguous semidirect product(C2×C7×C17)⋊(C3×C23)
unambiguous semidirect product(C2×C7×C23)⋊(C3×C17)
unambiguous semidirect product(C2×C7×C17×C23)⋊C3

Splittings

Fitting, socle, HallC2×C7×C17×C23 ⋊ C3
derived, SylowC7 ⋊ C2×C3×C17×C23
HallC7×C17×C23 ⋊ C2×C3
HallC2×C7×C23 ⋊ C3×C17
HallC2×C7×C17 ⋊ C3×C23
HallC7×C23 ⋊ C2×C3×C17
HallC7×C17 ⋊ C2×C3×C23
HallC2×C7 ⋊ C3×C17×C23

Subgroups

subgroupG/H
center, hypercenterC2×C17×C23ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C2×C3×C17×C23
FrattiniC1G
Fitting, socleC2×C7×C17×C23C3
LCSG ▷ C7
UCSC1 ◁ C2×C17×C23

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5474

Conjugacy classes & character table

3910 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC4×C17×ΓL(1,𝔽8)×C23, C17×ΓL(1,𝔽8)×Dic23, ΓL(1,𝔽8)×C23×Dic17, …
maximal normal subgroup ofC4×C17×ΓL(1,𝔽8)×C23, C17×ΓL(1,𝔽8)×Dic23, ΓL(1,𝔽8)×C23×Dic17, …
maximal quotient ofC4×C17×ΓL(1,𝔽8)×C23, C22×C17×ΓL(1,𝔽8)×C23

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16422, 2);