Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10620198 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C32×C4 |
No aliases for this group.
| derived, Sylow | C34 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620198 |
| commutator, hypocenter | C34 | C42 |
| Fitting, socle | C2×C34 | C2×C4 |
| prime radical, cosocle | C2×C3⋊(C3⋊(C3⋊D3)) | C22 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C2 | |
452 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3^4|a^2=c,b^2=d,e^a=ef,f^a=ef^2,g^a=gh,h^a=gh^2,e^b=e^2,f^b=f^2,e^c=e^2,f^c=f^2,g^c=g^2,h^c=h^248 conjugacy classes — view the full character table.
| derived subgroup of | #72252832 |
| maximal subgroup of | #72213324, #72213718, #72213723, … |
| maximal normal subgroup of | #72213324, #72213718, #72213723, … |
| maximal quotient of | C2×#13049409, #72220321, #72269678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3127);