Finite Group Databasebeta

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AGL(1,𝔽7)×#10624749 #72249156

order30618
automorphism?
homologyAbelianization (H1) = C2×C34 · Schur multiplier (H2) = C38

Aliases

direct productAGL(1,𝔽7)×#10624749

Splittings

FittingC7×#10624749 ⋊ C2×C3
HallΓL(1,𝔽8)×#10624749 ⋊ C2
HallD7 ⋊ C3×#10624749
SylowC7 ⋊ C2×C3×#10624749

Subgroups

subgroupG/H
centerC3AGL(1,𝔽7)×C32⋊H(3,𝔽3)
commutator, cosocleC33×C7C2×C34
FrattiniC33C33×AGL(1,𝔽7)
FittingC7×#10624749C2×C3
socleC3×C7C2×C3×C32⋊H(3,𝔽3)
hypercenter#10624749AGL(1,𝔽7)
hypocenterC7C2×C3×#10624749
LCSG ▷ C33×C7 ▷ C3×C7 ▷ C7
UCSC1 ◁ C3 ◁ C33 ◁ #10624749

9896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 189

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,𝔽7)×#13091120
maximal normal subgroup ofAGL(1,𝔽7)×#13091120

GAP

Past the SmallGroups library coverage — no SmallGroups id.