Finite Group DatabasebetaFinite Group Databasebeta| order | 27216 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C23×C3×C4×C81 (order 7776) · Inn(G) = C1 · Out(G) = C23×C3×C4×C81 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C27216 |
| abelian | C7×C16×C243 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C8×C81 | C2×C3×C7 |
| socle, prime radical | C2×C3×C7 | C8×C81 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
60 conjugacy classes of subgroups.
27216 conjugacy classes. The character table is too large to display.
| Fitting subgroup of | Dic13608, D27216 |
| maximal subgroup of | Dic13608, D27216 |
| maximal normal subgroup of | Dic13608, D27216 |
Past the SmallGroups library coverage — no SmallGroups id.