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AGL(1,ℤ/8)×#3861 #72245144

order4704
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productAGL(1,ℤ/8)×#3861

Splittings

Fitting, HallC72×AGL(1,ℤ/8) ⋊ C3
SylowC72 ⋊ C3×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC2C2×D4×#3861
commutator, cosocleC4×C72C23×C3
FrattiniC4C23×#3861
FittingC72×AGL(1,ℤ/8)C3
socleC2×C72C2×C3×D4
hypercenterAGL(1,ℤ/8)#3861
hypocenterC72C3×AGL(1,ℤ/8)
LCSG ▷ C4×C72 ▷ C2×C72 ▷ C72
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

680 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 392

Conjugacy classes & character table

209 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/8)×#64424, ΓL(1,𝔽8)2×AGL(1,ℤ/8), #453×#3861, …
maximal normal subgroup ofAGL(1,ℤ/8)×#64424, ΓL(1,𝔽8)2×AGL(1,ℤ/8), #453×#3861, …
maximal quotient of#442×#3861, C2×AGL(1,ℤ/8)×#3861

GAP

Past the SmallGroups library coverage — no SmallGroups id.