Finite Group Databasebeta

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C7×Ω+(4,𝔽3) #72246229

order2016
automorphism?
homologyAbelianization (H1) = C32×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC7×Ω+(4,𝔽3)

Splittings

Fitting, HallC7×H(5,𝔽2) ⋊ C32
derived, SylowH(5,𝔽2) ⋊ C32×C7

Subgroups

subgroupG/H
center, socle, hypercenterC2×C7A42
commutator, cosocle, hypocenterH(5,𝔽2)C32×C7
FrattiniC2C7×A42
FittingC7×H(5,𝔽2)C32
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C2×C7

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 168

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

derived subgroup ofΓL(1,𝔽8)×SO+(4,𝔽3), #72246255, #72246260, …
maximal subgroup ofΓL(1,𝔽8)×Ω+(4,𝔽3), D7×Ω+(4,𝔽3), C7×SO+(4,𝔽3), …
maximal normal subgroup ofΓL(1,𝔽8)×Ω+(4,𝔽3), D7×Ω+(4,𝔽3), C7×SO+(4,𝔽3), …

GAP

Past the SmallGroups library coverage — no SmallGroups id.