Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C35 |
| unambiguous semidirect product | C3⋊(C3⋊#1084) |
| derived, Sylow | C32×H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | C3⋊(C3⋊(C3⋊D3)) |
| commutator, hypocenter | C32×H(3,𝔽3) | C4 |
| Fitting, prime radical, cosocle | C2×C32×H(3,𝔽3) | C2 |
| socle | C2×C33 | C3⋊D3 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
1062 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^2,3^5|a^2=b,c^a=c^2,d^a=d^2,e^a=e^2,f^a=f^2,f^e=fg108 conjugacy classes — view the full character table.
| maximal subgroup of | #14518600, #14518773, ((C3⋊D3)⋊C3)⋊(C3⋊Dic3), … |
| maximal normal subgroup of | #14518600, #14518773, ((C3⋊D3)⋊C3)⋊(C3⋊Dic3), … |
| maximal quotient of | C3⋊(C3⋊#6180), C2×C3⋊(C3⋊#1084) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 749);