Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #14554330 |
| 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 | |
2024 conjugacy classes of subgroups; only the 311 normal subgroups are shown.
This group has 2024 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|c^3=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,e^b=eg,d^c=dg155 conjugacy classes — view the full character table.
| Fitting subgroup of | #72243639, #72243684, #72243847 |
| maximal subgroup of | #72221882, #72221946, #72243639, … |
| maximal normal subgroup of | #72221882, #72221946, #72243639, … |
| maximal quotient of | C2×#14554332 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 8191);