Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of C3⋊(C3⋊D27) |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊Dic27) |
| unambiguous semidirect product | C27⋊(C3⋊Dic3) |
| derived, Sylow | C32×C27 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊D27) |
| commutator, hypocenter | C32×C27 | C4 |
| Frattini | C2×C9 | C3⋊(C3⋊D3) |
| Fitting, cosocle | C2×C32×C27 | C2 |
| socle, prime radical | C2×C33 | D9 |
| LCS | G ▷ C32×C27 | |
| UCS | C1 ◁ C2 | |
216 conjugacy classes of subgroups.
| transitive degree | μt(G) = 972 |
Polycyclic presentation:
2^2,3^5|a^2=b,c^3=f,f^3=g,c^a=c^2f^2g^2,d^a=d^2,e^a=e^2,f^a=f^2g^2,g^a=g^2246 conjugacy classes — view the full character table.
| maximal subgroup of | #14516997, #14516998, C3⋊Dic3×D27, … |
| maximal normal subgroup of | #14516997, #14516998, C3⋊Dic3×D27, … |
| maximal quotient of | C3⋊(C3⋊(C27⋊C8)), C2×C3⋊(C3⋊Dic27) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 368);