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C2×C3⋊(ΓL(1,𝔽8)⋊D89) #36375896

order22428
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×C3⋊(ΓL(1,𝔽8)⋊D89)
unambiguous semidirect product(C2×C3)⋊(ΓL(1,𝔽8)⋊D89)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊D267
unambiguous semidirect product(C2×C89)⋊(C3⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×D1869)⋊C3

Splittings

Fitting, socleC2×C3×C7×C89 ⋊ C2×C3
derivedC3×C7×C89 ⋊ C22×C3
HallC3×ΓL(1,𝔽8)×C89 ⋊ C22
HallC7×C89 ⋊ C2×C3×D3
SylowC89 ⋊ C2×C3⋊AGL(1,𝔽7)
HallC3×ΓL(1,𝔽8) ⋊ C2×D89
SylowC7 ⋊ C2×C3×D267

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(ΓL(1,𝔽8)⋊D89)
commutator, cosocle, hypocenterC3×C7×C89C22×C3
FrattiniC1G
Fitting, socleC2×C3×C7×C89C2×C3
LCSG ▷ C3×C7×C89
UCSC1 ◁ C2

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 3738

Conjugacy classes & character table

1344 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22428, 78);