Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #4174⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| derived, Sylow | AΣL(1,𝔽27) ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #4174⋊C2 |
| commutator, hypocenter | AΣL(1,𝔽27) | C42 |
| Frattini | C22×C32 | D32 |
| Fitting, prime radical, cosocle | C22×AΣL(1,𝔽27) | C22 |
| socle | C22×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C22 | |
256 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 46 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:
2^4,3^4|a^2=d,b^2=c,f^a=f^2,g^a=g^2,h^a=h^2,e^b=e^2,g^b=g^2h,f^e=fg,g^e=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 198);