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#13135751

order1728
automorphism?
homologyAbelianization (H1) = C4×C8 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

SylowH(3,𝔽3) ⋊ #346

Subgroups

subgroupG/H
center, FrattiniC2×C3×C8C2×C3⋊D3
commutatorC2×H(3,𝔽3)C4×C8
FittingC4×C8×H(3,𝔽3)C2
socleC22×C3C2×C3⋊(C3⋊C8)
prime radicalC22×H(3,𝔽3)C2×C8
cosocleC2×C8×H(3,𝔽3)C22
hypercenterC3×C4×C8C3⋊D3
hypocenterH(3,𝔽3)#346
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C8 ◁ C3×C4×C8

297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 576

Polycyclic presentation:

2^6,3^3|a^2=d,b^2=c,d^2=e,e^2=f,b^a=bf,g^a=g^2,h^a=h^2,h^g=hi

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 1814);