Finite Group Databasebeta

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ΓL(1,𝔽8)×C23×D33 #44541802

order31878
automorphism?
homologyAbelianization (H1) = C2×C3×C23 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×C23×D33
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×D11×C23)
unambiguous semidirect productC11⋊(D3×ΓL(1,𝔽8)×C23)
unambiguous semidirect product(C7×D33)⋊(C3×C23)
unambiguous semidirect product(C7×C23×D33)⋊C3

Splittings

Fitting, socleC3×C7×C11×C23 ⋊ C2×C3
derivedC3×C7×C11 ⋊ C2×C3×C23
HallC3×C11×ΓL(1,𝔽8)×C23 ⋊ C2
HallC7×C11×C23 ⋊ C3×D3
HallC3×ΓL(1,𝔽8)×C23 ⋊ D11
HallC3×C11×ΓL(1,𝔽8) ⋊ C2×C23
HallC11×C23 ⋊ D3×ΓL(1,𝔽8)
HallC7×C23 ⋊ C3×D33
HallC7×C11 ⋊ C3×D3×C23
HallC3×ΓL(1,𝔽8) ⋊ D11×C23
SylowC11 ⋊ D3×ΓL(1,𝔽8)×C23
SylowC7 ⋊ C3×C23×D33

Subgroups

subgroupG/H
center, hypercenterC23ΓL(1,𝔽8)×D33
commutator, cosocle, hypocenterC3×C7×C11C2×C3×C23
FrattiniC1G
Fitting, socleC3×C7×C11×C23C2×C3
LCSG ▷ C3×C7×C11
UCSC1 ◁ C23

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 5313

Conjugacy classes & character table

2070 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31878, 71);