Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived | C32×Q8 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | (C3⋊D3)⋊A4 |
| commutator, hypocenter | C32×Q8 | C2×C9 |
| Frattini, socle | C2×C32 | D3×A4 |
| Fitting, cosocle | C33×Q8 | C2×C3 |
| prime radical | C2×C3×C3⋊D3 | A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C2×C3 | |
151 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 151 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^2,2^2,3^2,2|b^3=c,d^2=h,e^2=h,f^a=f^2,g^a=g^2,d^b=de,e^b=d,f^b=fg,e^d=eh78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 780);