Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
No aliases for this group.
| Fitting | C24×C32 ⋊ C2×D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C3⋊S4 |
| commutator, cosocle | C2×C32×A4 | C23 |
| Fitting | C24×C32 | C2×D3 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C22 | D3×C3⋊S4 |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1475 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 1475 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^4,3^3,2^2|c^a=cd,e^a=e^2,f^a=f^2,h^a=hi,c^b=cd,g^b=g^2,h^e=i,i^e=hi75 conjugacy classes — view the full character table.
| maximal subgroup of | #72222956, #72222975, S4×#3811, … |
| maximal normal subgroup of | S4×#3811, D3×D4×C3⋊S4, D3×#10596473 |
| maximal quotient of | #72222948, #72222973, #72252927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46905);