Finite Group Databasebeta

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ΓL(1,𝔽8)×#1438 #72230200

order2688
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C27

Aliases

direct productΓL(1,𝔽8)×#1438

Splittings

Fitting, HallC7×#1438 ⋊ C3
SylowC7 ⋊ C3×#1438

Subgroups

subgroupG/H
center, FrattiniC24C23×ΓL(1,𝔽8)
commutatorC23×C7C22×C3×C4
FittingC7×#1438C3
socle, cosocleC24×C7C23×C3
prime radicalC23×D4×ΓL(1,𝔽8)C2
hypercenter#1438ΓL(1,𝔽8)
hypocenterC7C3×#1438
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1438

2336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 448

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72276889, AGL(1,𝔽7)×#1438, ΓL(1,𝔽8)×#15947
maximal normal subgroup of#72276889, AGL(1,𝔽7)×#1438, ΓL(1,𝔽8)×#15947

GAP

Past the SmallGroups library coverage — no SmallGroups id.