Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #66432 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived, Sylow | C3×H(3,𝔽3) ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #66432 |
| commutator, hypocenter | C3×H(3,𝔽3) | C2×C8 |
| Frattini | C3×C4 | #1113 |
| Fitting, cosocle | C3×C4×H(3,𝔽3) | C22 |
| socle | C2×C32 | #624 |
| prime radical | C2×C3×#272 | C4 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C4 | |
301 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^4,3^4|b^2=c,c^2=d,e^a=e^2,g^a=g^2,e^b=e^2,f^b=f^2,h^b=h^2,g^e=gh84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1465);