Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #13049802 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, Sylow | C34 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle | C3⋊(C3⋊#157) | C22 |
| Fitting, socle, hypocenter | C34 | SD16 |
| prime radical | #10620203 | C2 |
| LCS | G ▷ C3⋊(C3⋊#157) ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 | |
116 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3^4|b^2=d,c^2=d,b^a=bcd,c^a=cd,e^a=ef,f^a=f^2,g^a=g^2,c^b=cd,f^b=ef^2,g^b=gh^2,h^b=g^2h^2,e^c=e^2,f^c=f^2,g^c=h^2,h^c=g,g^d=g^2,h^d=h^224 conjugacy classes — view the full character table.
| maximal subgroup of | #72220425, #72277538, C3×#13049802, … |
| maximal normal subgroup of | #72220425, C3×#13049802 |
| maximal quotient of | #72277511, #72277538, C3×#13049802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3520);