Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11724202 |
| commutator, cosocle | C32×#41⋊C3 | C22 |
| Frattini, hypercenter | C4 | D3×C3⋊S4 |
| Fitting | C32×#41 | C2×D3 |
| socle | C2×C32 | C4⋊S4 |
| hypocenter | C32×SL(2,𝔽3) | D4 |
| LCS | G ▷ C32×#41⋊C3 ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
456 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=ci,c^2=i,g^2=i,h^2=i,b^a=bc,c^a=ci,d^a=d^2,e^a=e^2,g^a=gh,h^a=hi,f^b=f^2,g^d=h,h^d=ghi,h^g=hi75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32112);