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#67×AGL(1,ℤ/8) #11670774

order768
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct product#67×AGL(1,ℤ/8)

Splittings

FittingC22×C3×AGL(1,ℤ/8) ⋊ C2
SylowC3 ⋊ D4×AGL(1,ℤ/8)

Subgroups

subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC22×C3×AGL(1,ℤ/8)C2
socleC22×C3C23×D4
hypercenterC22×AGL(1,ℤ/8)D3
hypocenterC3D4×AGL(1,ℤ/8)
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×AGL(1,ℤ/8)

3230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 96

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×D4×AGL(1,ℤ/8), #72253850, #72253889, …
maximal normal subgroup ofD3×D4×AGL(1,ℤ/8), #72253850, #72253889, …
maximal quotient ofAGL(1,ℤ/8)×GL(2,ℤ/4)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1045155);