Finite Group Databasebeta

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D4×(C3⋊(C3⋊C8))⋊C3 #13142738

order1728
automorphism?
homologyAbelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C23×C3

Aliases

direct productD4×(C3⋊(C3⋊C8))⋊C3
unambiguous semidirect product(D4×C3⋊(C3⋊C8))⋊C3

Splittings

SylowH(3,𝔽3) ⋊ C8×D4

Subgroups

subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC2×C32C22×C3×C8
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C22×C3×Dic3
prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×C32×C4C23×C3
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×C8×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4×D4

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 288

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 8801);