Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #4178⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4178⋊C8 |
| derived, Sylow | #732 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4178⋊C2 |
| commutator, hypocenter | #732 | C2×C8 |
| Frattini | C32×C4 | D32 |
| Fitting, cosocle | C4×#732 | C22 |
| socle | C2×C3 | ((C3⋊D3)⋊C3)⋊C4 |
| prime radical | C2×#4178 | C4 |
| LCS | G ▷ #732 | |
| UCS | C1 ◁ C4 | |
122 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^4,3^4|b^2=c,c^2=d,e^3=h,f^a=f^2,g^a=g^2h,e^b=e^2h^2,g^b=g^2,h^b=h^2,f^e=fg,g^f=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 162);