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C3×ΓL(1,𝔽8)×D23 #15668477

order2898
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)×D23
unambiguous semidirect productC23⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D23)⋊C32
unambiguous semidirect product(C3×C7×D23)⋊C3

Splittings

Fitting, socleC3×C7×C23 ⋊ C2×C3
derived, HallC7×C23 ⋊ C2×C32
HallC3×ΓL(1,𝔽8)×C23 ⋊ C2
HallC7×D23 ⋊ C32
SylowC23 ⋊ C2×C3×ΓL(1,𝔽8)
SylowC7 ⋊ C32×D23

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)×D23
commutator, cosocle, hypocenterC7×C23C2×C32
FrattiniC1G
Fitting, socleC3×C7×C23C2×C3
LCSG ▷ C7×C23
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 483

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×C5×ΓL(1,𝔽8)×D23, C3×ΓL(1,𝔽8)×D115, C3×D23×C49⋊C3, …
maximal normal subgroup ofC3×C5×ΓL(1,𝔽8)×D23, C3×C7×ΓL(1,𝔽8)×D23, C3×ΓL(1,𝔽8)×AGL(1,𝔽23), …
maximal quotient ofC3×C5×ΓL(1,𝔽8)×D23, C3×ΓL(1,𝔽8)×D115, C3×D23×C49⋊C3, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2898, 16);