Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3)2 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C34 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C33 | C34 |
| commutator | C32 | C33×C9 |
| prime radical | C34 | C33 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
974 conjugacy classes of subgroups; only the 494 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 494 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 729 |
Polycyclic presentation:
3^7|a^3=e,b^3=g,c^3=f^2,d^3=g,b^a=be,d^c=df363 conjugacy classes — view the full character table.
| Fitting subgroup of | #72243521, #72243543, #72243581 |
| maximal subgroup of | #72243517, #72243521, #72243543 |
| maximal normal subgroup of | #72243517, #72243521, #72243543 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 5948);