Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×C32 ⋊ C2×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#89101 |
| commutator, cosocle | C2×C32×A4 | C23 |
| Fitting | C24×C32 | C2×D3 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C22 | #89101 |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1118 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^3,2^2|e^a=e^2,f^a=f^2,h^a=hi,c^b=cd,f^b=f^2,g^b=g^2,h^e=i,i^e=hi75 conjugacy classes — view the full character table.
| maximal subgroup of | #72241491, D3×#67×S4, D3×#10596473, … |
| maximal normal subgroup of | D3×#67×S4, D3×#10596473, D4×#89101 |
| maximal quotient of | #72253005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46941);