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, Frattini, hypercenter | C2 | #10620185 |
| commutator, prime radical, cosocle, hypocenter | SL(2,ℤ/9) | C2 |
| Fitting, socle | C2×C33 | S4 |
| LCS | G ▷ SL(2,ℤ/9) | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2,3,2^3,3^3|a^2=e,c^2=e,d^2=e,b^a=b^2,c^a=de,d^a=ce,f^a=h,g^a=fgh^2,h^a=f,c^b=d,d^b=cde,f^b=h,g^b=f^2g^2h,h^b=g^2h,d^c=de,g^c=g^2,h^c=h^2,f^d=f^2,g^d=g^2h^229 conjugacy classes — view the full character table.
| maximal subgroup of | #72228152 |
| maximal quotient of | #72228152 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3084);