Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | H(3,𝔽3)×#95 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×(C3⋊D3)⋊C3 |
| commutator, cosocle | C23×C32 | C23×C3 |
| Frattini, socle | C23×C3 | C22×C3×D3 |
| Fitting | H(3,𝔽3)×#95 | C2 |
| hypercenter | #95 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×#392 |
| LCS | G ▷ C23×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ #95 | |
1203 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 1203 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|b^2=g,c^2=f,b^a=bg,c^a=cf,h^a=h^2,i^a=i^2,c^b=ce,h^d=hi146 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7422);