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S4×AGL(1,ℤ/8) #11714925

order768
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C26

Aliases

direct productS4×AGL(1,ℤ/8)

Splittings

FittingC22×AGL(1,ℤ/8) ⋊ D3

Subgroups

subgroupG/H
centerC2C2×D4×S4
commutator, cosocleC4×A4C24
FrattiniC4C23×S4
FittingC22×AGL(1,ℤ/8)D3
socleC23C2×D3×D4
hypercenterAGL(1,ℤ/8)S4
hypocenterA4C2×AGL(1,ℤ/8)
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

1649 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 1649 conjugacy classes of subgroups — too many to draw. Only its 121 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) = 24

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×S4×AGL(1,ℤ/8), #72228321, #72228331, …
maximal normal subgroup ofC2×S4×AGL(1,ℤ/8), S4×#453, #72258607, …
maximal quotient ofAGL(1,ℤ/8)×GL(2,𝔽3), C2×S4×AGL(1,ℤ/8), #72228321, …
automorphism group of#576, #578, #5398, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1089306);