Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C35 · Schur multiplier (H2) = C38 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C32 | C35 |
| prime radical | C33×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
5444 conjugacy classes of subgroups; only the 2861 normal subgroups are shown.
This group has 5444 conjugacy classes of subgroups — too many to draw. Only its 2861 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=g,b^a=bf,c^a=cg,e^d=ef315 conjugacy classes — view the full character table.
| Fitting subgroup of | #72243421, #72243456, #72243480, … |
| maximal subgroup of | #72243421, #72243456, #72243480 |
| maximal normal subgroup of | #72243421, #72243456, #72243480 |
| maximal quotient of | #72243407 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9141);