Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #14554315 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C38 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C3×C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C34 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C34 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
2216 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 2216 conjugacy classes of subgroups — too many to draw. Only its 311 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=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^b=fg,e^c=eg155 conjugacy classes — view the full character table.
| Fitting subgroup of | #72277665, #72277728 |
| maximal subgroup of | #72265004, #72265015, #72265023, … |
| maximal normal subgroup of | #72265004, #72265015, #72265023, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 8175);