Finite Group Databasebeta

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C3×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #72222172

order2646
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C33

Aliases

direct productC3×ΓL(1,𝔽8)⋊AGL(1,𝔽7)

Splittings

Fitting, socleC3×C72 ⋊ C2×C32
derived, SylowC72 ⋊ C2×C33
HallC3×ΓL(1,𝔽8)2 ⋊ C2
HallC7⋊D7 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC72C2×C33
FrattiniC1G
Fitting, socleC3×C72C2×C32
LCSG ▷ C72
UCSC1 ◁ C3

248 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), D3×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72224660, …
maximal normal subgroup ofD3×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72224660, C3×AGL(1,𝔽7)2, …
maximal quotient ofA4×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72224657, #72224660, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.