Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | #6856 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #4179 |
| commutator, hypocenter | C32⋊C9 | C3×C4 |
| Frattini | C2×C33 | C3×D3 |
| Fitting, prime radical | C2×#6856 | C2 |
| socle | C2×C32 | (C3⋊D3)⋊C3 |
| cosocle | C2×C32⋊C9 | C2×C3 |
| LCS | G ▷ C32⋊C9 | |
| UCS | C1 ◁ C2×C3 | |
77 conjugacy classes of subgroups.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2,3,2,3^4|a^2=c,b^3=g,d^3=f,d^a=d^2f^2,e^a=e^2g,f^a=f^2,d^b=de,e^b=ef^2,e^d=eg62 conjugacy classes — view the full character table.
| maximal subgroup of | #14515721, #92160⋊C4, #14515725, … |
| maximal normal subgroup of | #14515721, #92160⋊C4, #14515725, … |
| maximal quotient of | #14515620, C2×#11757283 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 8);