Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #14547270 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C36 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C32⋊H(3,𝔽3) |
| commutator, Frattini, cosocle | C34 | C33 |
| LCS | G ▷ C34 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C1 | |
800 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 800 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^7|a^3=fg,b^a=bd,c^a=ce,d^a=df,e^a=ef,d^b=dg,e^b=ef,d^c=df91 conjugacy classes — view the full character table.
| maximal subgroup of | #72264866, #72265344, #72265348, … |
| maximal normal subgroup of | #72264866, #72265344, #72265348, … |
| maximal quotient of | #72265348, #72265352, #72265356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 1136);