Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C24×C32 |
No aliases for this group.
| Fitting | C24×3−1+2 ⋊ C3 |
| Sylow | C24 ⋊ #737 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×C22⋊A4 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×3−1+2 | C3 |
| prime radical | C32⋊(C22⋊A4) | C3 |
| hypercenter | 3−1+2 | C22⋊A4 |
| hypocenter | C24 | #737 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
391 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
3^4,2^4|b^3=d,c^a=cd,e^a=ef,f^a=e,g^a=gh,h^a=g,c^b=cd88 conjugacy classes — view the full character table.
| maximal subgroup of | #72259653, #72259666, #72259686 |
| maximal normal subgroup of | #72259653, #72259666, #72259686 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3058);