Finite Group Databasebeta

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C4⋊C8×(C3⋊D3)⋊C3 #13142054

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

Aliases

direct productC4⋊C8×(C3⋊D3)⋊C3
unambiguous semidirect product(C3⋊D3×C4⋊C8)⋊C3

Splittings

FittingH(3,𝔽3)×C4⋊C8 ⋊ C2
SylowH(3,𝔽3) ⋊ C2×C4⋊C8

Subgroups

subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC2×C32C22×C3×C8
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

521 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, 8117);