Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #4174⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #4183⋊C8 |
| derived, Sylow | AΣL(1,𝔽27) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #4174⋊C2 |
| commutator, hypocenter | AΣL(1,𝔽27) | C2×C8 |
| Frattini | C32×C4 | D32 |
| Fitting, cosocle | C4×AΣL(1,𝔽27) | C22 |
| socle | C2×C3 | #6199 |
| prime radical | C2×#4183 | C4 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C4 | |
192 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 192 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) = 72 |
Polycyclic presentation:
2^4,3^4|a^2=c,b^2=c,c^2=d,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, 125);