Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #10624484 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | #6857 |
| commutator, Frattini, cosocle | C32×C9 | C33 |
| socle | C3 | C3×#6857 |
| prime radical | C3×#6908 | C3 |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ C32×C9 ◁ C33×C9 ◁ G | |
| LECS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
413 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 59 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|a^3=e,c^3=g,d^3=g^2,b^a=bd,d^a=de,f^a=fg,d^b=df,e^b=eg,f^b=fg123 conjugacy classes — view the full character table.
| Fitting subgroup of | #72277433 |
| maximal subgroup of | #72277433, C2×#14550839 |
| maximal normal subgroup of | #72277433, C2×#14550839 |
| maximal quotient of | C2×#14550839 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 4698);