Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C9⋊#1084 |
| derived, Sylow | C9×H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(C3⋊D9) |
| commutator, hypocenter | C9×H(3,𝔽3) | C4 |
| Frattini, socle | C2×C32 | C3⋊(C3⋊D3) |
| Fitting, cosocle | C2×C9×H(3,𝔽3) | C2 |
| prime radical | C2×C3×H(3,𝔽3) | D3 |
| LCS | G ▷ C9×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
263 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3^5|a^2=b,c^3=f,c^a=c^2f^2,d^a=d^2,e^a=e^2,f^a=f^2,e^d=eg108 conjugacy classes — view the full character table.
| maximal subgroup of | ((C3⋊D3)⋊C3)⋊Dic9, #14516971, #14516973, … |
| maximal normal subgroup of | ((C3⋊D3)⋊C3)⋊Dic9, #14516971, #14516973, … |
| maximal quotient of | C9⋊#6180, C2×C9⋊#1084 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 366);