Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C32×C9 ⋊ C22 |
| Sylow | C32×C9 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#66431 |
| commutator, cosocle | C2×C32×C9 | C23 |
| Frattini | C2×C3 | C2×#1113 |
| Fitting | C22×C32×C9 | C22 |
| socle | C2×C33 | C22×D3 |
| hypercenter | C22 | #66431 |
| hypocenter | C32×C9 | C2×D4 |
| LCS | G ▷ C2×C32×C9 ▷ C32×C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
464 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 66 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|f^3=h,e^a=e^2,f^a=f^2h^2,h^a=h^2,c^b=cd,e^b=e^2,g^b=g^2111 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#88668, #72259266, #72259536 |
| maximal normal subgroup of | D3×#88668, #72259536 |
| maximal quotient of | #72259266 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2489);