Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C34 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10620197 |
| commutator, Fitting, socle | C2×C34 | C2×C4 |
| prime radical | C2×C3⋊D32 | C2 |
| cosocle | C2×C32×C3⋊D3 | C22 |
| hypocenter | C34 | C22⋊C4 |
| LCS | G ▷ C2×C34 ▷ C34 | |
| UCS | C1 ◁ C2 | |
580 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3^4|b^2=c,b^a=bd,g^a=g^2,h^a=h^2,e^b=ef,f^b=ef^2,g^b=g^2,e^c=e^2,f^c=f^254 conjugacy classes — view the full character table.
| maximal subgroup of | #72220286, #72220703, #72220712, … |
| maximal normal subgroup of | #72220286, #72220703, #72220712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3121);