Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×S4 |
| commutator, cosocle | C2×C32×A4 | C23 |
| Fitting, socle | C23×C32 | C22×D3 |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
1114 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 64 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|b^2=d,c^2=d,b^a=bd,c^a=cd,f^a=f^2,g^b=g^2,e^c=e^2,h^c=hi,h^e=i,i^e=hi69 conjugacy classes — view the full character table.
| maximal subgroup of | #72222977, #72222984, #72222989, … |
| maximal normal subgroup of | #72222977, D3×#10596473 |
| maximal quotient of | #72253015 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46356);