Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #14555221 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C36 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C3×#6911 |
| commutator, Frattini, cosocle | C3×C9 | C34 |
| LCS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C32×3−1+2 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C32 ▷ C3 ▷ C1 | |
748 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 748 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^7|b^3=f^2,e^3=g^2,b^a=be,e^a=ef,f^a=fg,c^b=cf,d^b=dg,e^c=eg^2107 conjugacy classes — view the full character table.
| Fitting subgroup of | #72277466, #72277487, #72277494, … |
| maximal subgroup of | #72222026, #72277466, #72277487, … |
| maximal normal subgroup of | #72222026, #72277466, #72277487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9080);