Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C3×C4 |
No aliases for this group.
| derived, Sylow | C3×H(3,𝔽3) ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | ((C3⋊D3)⋊C3)⋊D3 |
| commutator, hypocenter | C3×H(3,𝔽3) | C42 |
| Frattini | C22×C3 | D3×C3⋊D3 |
| Fitting, prime radical, cosocle | C22×C3×H(3,𝔽3) | C22 |
| socle | C22×C32 | D32 |
| LCS | G ▷ C3×H(3,𝔽3) | |
| UCS | C1 ◁ C22 | |
433 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 433 conjugacy classes of subgroups — too many to draw. Only its 98 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,g^a=g^2,h^a=h^2,e^b=e^2,f^b=f^2,h^b=h^2,g^f=gh84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1435);