Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C32×C9 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D32×D9 |
| commutator, Fitting, cosocle | C2×C32×C9 | C23 |
| Frattini | C2×C3 | D33 |
| socle | C2×C33 | C22×D3 |
| hypocenter | C32×C9 | C2×D4 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 | |
496 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^4|c^2=d,f^3=h,b^a=bd,c^a=cd,g^a=g^2,c^b=cd,e^b=e^2,f^c=f^2h^2,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#88668, #72259423, #72259642 |
| maximal normal subgroup of | D3×#88668, #72259642 |
| maximal quotient of | #72259423 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2019);