Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C34 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620203 |
| commutator, cosocle | C32×C3⋊Dic3 | C22 |
| Fitting, socle | C2×C34 | D4 |
| hypocenter | C34 | D8 |
| LCS | G ▷ C32×C3⋊Dic3 ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
290 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3^4|c^2=d,b^a=bc,c^a=cd,e^a=e^2,h^a=h^2,c^b=cd,e^b=f,f^b=e,g^b=g^2,e^c=e^2,f^c=f^257 conjugacy classes — view the full character table.
| maximal subgroup of | #72220232, #72220276, #72220277 |
| maximal normal subgroup of | #72220232, #72220276, #72220277 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3158);