Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of D27⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #6864⋊C4 |
| derived | C27 ⋊ C4×C9 |
| Sylow | #6864 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D27⋊C3 |
| commutator, hypocenter | C27 | C4×C9 |
| Frattini | C2×C3×C9 | C3×D3 |
| Fitting | C2×#6864 | C2 |
| socle, prime radical | C2×C32 | C3×D9 |
| cosocle | C2×C3×C27 | C2×C3 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C2×C3 | |
54 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3,2,3^4|a^2=c,b^3=d,e^3=f,f^3=g,e^a=e^2f^2g^2,f^a=f^2g^2,g^a=g^2,e^b=eg126 conjugacy classes — view the full character table.
| maximal subgroup of | #6864⋊Q8, C4×#6864⋊C2, C2×#6864⋊C4, … |
| maximal normal subgroup of | #6864⋊Q8, C4×#6864⋊C2, C2×#6864⋊C4, … |
| maximal quotient of | #6864⋊C8, C2×#6864⋊C4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 18);