Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×C33 ⋊ C22 |
| Sylow | C33 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11723729 |
| commutator, cosocle | C23×C33 | C23 |
| Frattini | C23 | C2×#1113 |
| Fitting | C24×C33 | C22 |
| socle | C2×C33 | #120 |
| hypercenter | C24 | #1113 |
| hypocenter | C33 | B(4,𝔽2) |
| LCS | G ▷ C23×C33 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 | |
1316 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^6,3^3|c^a=ce,d^a=df,g^a=g^2,h^a=h^2,c^b=cd,e^b=ef,g^b=g^2,i^b=i^2138 conjugacy classes — view the full character table.
| maximal subgroup of | #72220714, D3×#10591725, #72220923, … |
| maximal normal subgroup of | D3×#10591725, #72220923, #72220980 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 28127);