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C2×C22⋊C4×(C3⋊D3)⋊C3 #13152360

order1728
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27×C3

Aliases

direct productC2×C22⋊C4×(C3⋊D3)⋊C3
unambiguous semidirect product(C2×C22⋊C4×C3⋊D3)⋊C3

Splittings

FittingC2×C22⋊C4×H(3,𝔽3) ⋊ C2
SylowH(3,𝔽3) ⋊ C22×C22⋊C4

Subgroups

subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
prime radicalC24×(C3⋊D3)⋊C3C2
cosocleC22×C32C24×C3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C22×C3×C22⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

2491 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

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, 18423);