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×C3 | C9⋊S4 |
| commutator, cosocle, hypocenter | (C3×SL(2,𝔽3))⋊C9 | C2 |
| Frattini, socle | C2×C32 | C3⋊S4 |
| Fitting | C3×Q8×C9 | D3 |
| prime radical | C32×SL(2,𝔽3) | D3 |
| LCS | G ▷ (C3×SL(2,𝔽3))⋊C9 | |
| UCS | C1 ◁ C2×C3 | |
121 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^4,2^3|a^2=h,c^3=e,f^2=h,g^2=h,b^a=b^2,c^a=c^2e^2,e^a=e^2,f^a=fg,g^a=gh,c^b=cd,f^b=g,g^b=fgh,g^f=gh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 713);