Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C22⋊C81)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C81 | C2 |
| Frattini | C2×C27 | S4 |
| Fitting | Q8×C27 | D3 |
| socle, prime radical | C2×C3 | (C22⋊C27)⋊C2 |
| LCS | G ▷ Q8⋊C81 | |
| UCS | C1 ◁ C2 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 1296 |
Polycyclic presentation:
2,3^4,2^3|a^2=h,b^3=c,c^3=d,d^3=e,f^2=h,g^2=h,b^a=b^2c^2d^2e^2,c^a=c^2d^2e^2,d^a=d^2e^2,e^a=e^2,f^a=fg,g^a=gh,f^b=g,g^b=fgh,g^f=gh99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 57);