Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4⋊AGL(1,ℤ/9) |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C2×C3 |
| Frattini, socle | C2×C3 | C3×C3⋊S4 |
| Fitting | Q8×3−1+2 | D3 |
| prime radical | C32×SL(2,𝔽3) | D3 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
146 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^4,2^3|a^2=h,d^3=e,f^2=h,g^2=h,c^a=c^2,d^a=d^2e^2,e^a=e^2,f^a=fg,g^a=gh,d^b=de^2,f^c=g,g^c=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #72245023, #72245682, #72245766, … |
| maximal normal subgroup of | #72245682, #72245766 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2108);