Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C33 ⋊ B(4,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722950 |
| commutator, Fitting, cosocle | C23×C33 | C23 |
| Frattini, hypercenter | C23 | D33 |
| socle | C2×C33 | #120 |
| hypocenter | C33 | B(4,𝔽2) |
| LCS | G ▷ C23×C33 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C23 | |
1194 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 1194 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|b^2=e,c^2=de,b^a=be,c^a=ce,d^a=df,i^a=i^2,c^b=cd,d^b=df,g^b=g^2,d^c=df,e^c=ef,h^c=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | #72220954, #72258283, D3×#10590410 |
| maximal normal subgroup of | #72220954, #72258283, D3×#10590410 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23052);