Finite Group Databasebeta

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AGL(1,𝔽7)×#3811 #72222058

order6048
automorphism?
homologyAbelianization (H1) = C24×C3 · Schur multiplier (H2) = C26

Aliases

direct productAGL(1,𝔽7)×#3811

Splittings

FittingC22×C32×C7 ⋊ C23×C3
HallC32×ΓL(1,𝔽8) ⋊ C22×D4
SylowC7 ⋊ C2×C3×#3811

Subgroups

subgroupG/H
center, FrattiniC2C2×D32×AGL(1,𝔽7)
commutator, socle, cosocleC2×C32×C7C24×C3
FittingC22×C32×C7C23×C3
hypercenterC22D32×AGL(1,𝔽7)
hypocenterC32×C7C22×C3×D4
LCSG ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C22

2436 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72222273, C2×AGL(1,𝔽7)×#3811, D32×D4×AGL(1,𝔽7), …
maximal normal subgroup ofC2×AGL(1,𝔽7)×#3811, D32×D4×AGL(1,𝔽7), AGL(1,𝔽7)×#64312, …
maximal quotient of#72222273, C2×AGL(1,𝔽7)×#3811, AGL(1,𝔽7)×#63930, …
holomorph ofC7×Dic3, C2×D3×C7

GAP

Past the SmallGroups library coverage — no SmallGroups id.