Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C8 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C82 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, hypocenter | H(3,𝔽3) | C82 |
| Frattini | C3×C42 | D32 |
| Fitting, cosocle | C42×H(3,𝔽3) | C22 |
| socle | C22×C3 | Dic32 |
| prime radical | C22×H(3,𝔽3) | C42 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C42 | |
303 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^3|a^2=d,b^2=c,c^2=e,d^2=f,h^a=h^2,i^a=i^2,g^b=g^2,i^b=i^2,h^g=hi176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1023);