Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C9 |
No aliases for this group.
| Fitting | C8×C92 ⋊ C2 |
| Sylow | C92 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊(C9⋊D9) |
| commutator, cosocle | C4×C92 | C22 |
| Frattini | C32×C4 | C2×C3⋊D3 |
| Fitting | C8×C92 | C2 |
| socle | C2×C32 | C4⋊(C3⋊D3) |
| hypercenter | C8 | C9⋊D9 |
| hypocenter | C92 | D8 |
| LCS | G ▷ C4×C92 ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
253 conjugacy classes of subgroups.
| transitive degree | μt(G) = 648 |
Polycyclic presentation:
2^4,3^4|b^2=cd,c^2=d,e^3=h,f^3=g,b^a=bc,c^a=cd,e^a=e^2h^2,f^a=f^2g^2,g^a=g^2,h^a=h^2327 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13046732, D8×C9⋊D9, #72244189, … |
| maximal normal subgroup of | C2×#13046732, D8×C9⋊D9, #72244189, … |
| maximal quotient of | C2×#13046732, #72244232 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 450);