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AGL(1,ℤ/8)×#272 #13154381

order1728
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

direct productAGL(1,ℤ/8)×#272

Splittings

FittingH(3,𝔽3)×AGL(1,ℤ/8) ⋊ C2
SylowH(3,𝔽3) ⋊ C2×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC2×C3C2×D4×C3⋊D3
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C23×C3⋊D3
FittingH(3,𝔽3)×AGL(1,ℤ/8)C2
hypercenterC3×AGL(1,ℤ/8)C3⋊D3
hypocenterH(3,𝔽3)C2×AGL(1,ℤ/8)
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×AGL(1,ℤ/8)

1639 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,ℤ/8)×((C3⋊D3)⋊C3)⋊C2
maximal normal subgroup ofAGL(1,ℤ/8)×((C3⋊D3)⋊C3)⋊C2
automorphism group ofD8×3−1+2, Q16×3−1+2

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 20444);