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C17×ΓL(1,𝔽8)×D23 #30655993

order16422
automorphism?
homologyAbelianization (H1) = C2×C3×C17 · Schur multiplier (H2) = C1

Aliases

direct productC17×ΓL(1,𝔽8)×D23
unambiguous semidirect productC23⋊(C2×C17×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D23)⋊(C3×C17)
unambiguous semidirect product(C7×C17×D23)⋊C3

Splittings

Fitting, socle, HallC7×C17×C23 ⋊ C2×C3
derived, HallC7×C23 ⋊ C2×C3×C17
HallC17×ΓL(1,𝔽8)×C23 ⋊ C2
HallC7×C17×D23 ⋊ C3
HallΓL(1,𝔽8)×C23 ⋊ C2×C17
HallC17×C23 ⋊ C2×ΓL(1,𝔽8)
HallC7×D23 ⋊ C3×C17
HallC7×C17 ⋊ C3×D23
SylowC23 ⋊ C2×C17×ΓL(1,𝔽8)
SylowC7 ⋊ C3×C17×D23

Subgroups

subgroupG/H
center, hypercenterC17ΓL(1,𝔽8)×D23
commutator, cosocle, hypocenterC7×C23C2×C3×C17
FrattiniC1G
Fitting, socleC7×C17×C23C2×C3
LCSG ▷ C7×C23
UCSC1 ◁ C17

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2737

Conjugacy classes & character table

1105 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC17×AGL(1,𝔽7)×D23, D23×C17⋊AGL(1,𝔽7), ΓL(1,𝔽8)×D17×D23, …
maximal normal subgroup ofC17×AGL(1,𝔽7)×D23, D23×C17⋊AGL(1,𝔽7), ΓL(1,𝔽8)×D17×D23, …
maximal quotient ofC17×ΓL(1,𝔽8)×Dic23, C2×C17×ΓL(1,𝔽8)×D23

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16422, 18);