Finite Group Databasebeta

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#13144415

order1728
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

SylowH(3,𝔽3) ⋊ #435

Subgroups

subgroupG/H
center, FrattiniC2×C3×C4C22×C3⋊D3
commutatorC22×H(3,𝔽3)C22×C4
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C22×C3⋊Dic3
prime radicalC2×D4×H(3,𝔽3)C4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC3×C4×D4C3⋊D3
hypocenterH(3,𝔽3)#435
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×D4

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 288

Polycyclic presentation:

2^6,3^3|a^2=e,e^2=f,c^a=cf,g^a=g^2,h^a=h^2,c^b=cd,h^g=hi

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 10478);