Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C22×C32 |
No aliases for this group.
| Fitting | C24×C9 ⋊ C32 |
| Sylow | C24 ⋊ C3×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×A42 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×C9 | C32 |
| prime radical | C3×A42 | C3 |
| hypercenter | C9 | A42 |
| hypocenter | C24 | C3×3−1+2 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C9 | |
217 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
3^4,2^4|c^3=d,c^a=cd,e^a=ef,f^a=e,c^b=cd,g^b=gh,h^b=g64 conjugacy classes — view the full character table.
| maximal subgroup of | #72228707, A42×3−1+2, #72252814, … |
| maximal normal subgroup of | #72228707, A42×3−1+2, #72259463, … |
| maximal quotient of | #72229281, #72252814, C2×#13049233, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2951);