Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #13049800 |
| 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 | C32×#157 | C22 |
| Fitting, socle, hypocenter | C34 | SD16 |
| prime radical | #157⋊(C3⋊D3) | C2 |
| LCS | G ▷ C32×#157 ▷ C32×C3⋊D3 ▷ C34 | |
| UCS | C1 | |
161 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 22 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=cd,c^2=d,b^a=bcd,c^a=cd,e^a=e^2f,g^a=g^2,h^a=h^2,e^b=ef,f^b=e,g^b=g^2,e^c=ef^2,f^c=e^2f^2,e^d=e^2,f^d=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#89092, D3×AGL(1,𝔽9)⋊D3, #72214303, … |
| maximal normal subgroup of | D3×#89092, D3×AGL(1,𝔽9)⋊D3, #72214303, … |
| maximal quotient of | C2×#13049800, #72259744 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3518);