Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of #92172 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C3×C27 ⋊ C3×C4 |
| Sylow | C32⋊C27 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #92172 |
| commutator, hypocenter | C3×C27 | C3×C4 |
| Frattini | C2×C3×C9 | C3×D3 |
| Fitting | C2×C32⋊C27 | C2 |
| socle | C2×C32 | C3×D9 |
| prime radical | C2×C33 | D9 |
| cosocle | C2×C3×C27 | C2×C3 |
| LCS | G ▷ C3×C27 | |
| UCS | C1 ◁ C2 | |
81 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
Polycyclic presentation:
2,3,2,3^4|a^2=c,d^3=f,f^3=g,d^a=d^2f^2g^2,e^a=e^2,f^a=f^2g^2,g^a=g^2,d^b=de108 conjugacy classes — view the full character table.
| maximal subgroup of | #14515763, #92206⋊C4, ((C3⋊D3)⋊C27)⋊C4, … |
| maximal normal subgroup of | #14515763, #92206⋊C4, ((C3⋊D3)⋊C27)⋊C4, … |
| maximal quotient of | #14515632, C2×#11757295 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 20);