Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of #4175 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C33 ⋊ C4×C9 |
| Sylow | #6866 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #4175 |
| commutator, hypocenter | C33 | C4×C9 |
| Frattini | C2×C33 | C3×D3 |
| Fitting | C2×#6866 | C2 |
| socle | C2×C32 | (C3⋊D3)⋊C3 |
| prime radical, cosocle | C2×C34 | C2×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
146 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3,2,3^4|a^2=c,b^3=d,e^a=e^2,f^a=f^2,g^a=g^2,e^b=ef,f^b=fg78 conjugacy classes — view the full character table.
| maximal subgroup of | #92207⋊C4, #92174⋊C4, #14515774, … |
| maximal normal subgroup of | #92207⋊C4, #92174⋊C4, #14515774, … |
| maximal quotient of | #14515633, C2×#11757296 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 21);