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ΓL(1,𝔽8)×D561 #37555646

order23562
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D561
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D187)
unambiguous semidirect productC11⋊(ΓL(1,𝔽8)×D51)
unambiguous semidirect productC17⋊(ΓL(1,𝔽8)×D33)
unambiguous semidirect product(C7×D561)⋊C3

Splittings

Fitting, socle, derivedC3×C7×C11×C17 ⋊ C2×C3
HallC3×C11×C17×ΓL(1,𝔽8) ⋊ C2
HallC7×C11×C17 ⋊ C3×D3
HallC3×C17×ΓL(1,𝔽8) ⋊ D11
HallC3×C11×ΓL(1,𝔽8) ⋊ D17
HallC11×C17 ⋊ D3×ΓL(1,𝔽8)
HallC7×C17 ⋊ C3×D33
HallC7×C11 ⋊ C3×D51
HallC3×ΓL(1,𝔽8) ⋊ D187
SylowC17 ⋊ ΓL(1,𝔽8)×D33
SylowC11 ⋊ ΓL(1,𝔽8)×D51
SylowC7 ⋊ C3×D561

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C7×C11×C17C2×C3
LCSG ▷ C3×C7×C11×C17
UCSC1

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3927

Conjugacy classes & character table

1410 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23562, 40);