Finite Group Databasebeta

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ΓL(1,𝔽8)×#10624749 #72249097

order15309
automorphism?
homologyAbelianization (H1) = C34 · Schur multiplier (H2) = C38

Aliases

direct productΓL(1,𝔽8)×#10624749

Splittings

FittingC7×#10624749 ⋊ C3
SylowC7 ⋊ C3×#10624749

Subgroups

subgroupG/H
centerC3ΓL(1,𝔽8)×C32⋊H(3,𝔽3)
commutator, cosocleC33×C7C34
FrattiniC33C33×ΓL(1,𝔽8)
FittingC7×#10624749C3
socleC3×C7C3×C32⋊H(3,𝔽3)
hypercenter#10624749ΓL(1,𝔽8)
hypocenterC7C3×#10624749
LCSG ▷ C33×C7 ▷ C3×C7 ▷ C7
UCSC1 ◁ C3 ◁ C33 ◁ #10624749

4948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 189

Conjugacy classes & character table

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

Relationships

maximal subgroup ofAGL(1,𝔽7)×#10624749, ΓL(1,𝔽8)×#13091120, #72254920
maximal normal subgroup ofAGL(1,𝔽7)×#10624749, ΓL(1,𝔽8)×#13091120, #72254920

GAP

Past the SmallGroups library coverage — no SmallGroups id.