Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | #731 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10619558 |
| commutator, Fitting, cosocle | C4×#731 | C22 |
| Frattini | C32×C4 | D32 |
| socle | C2×C3 | #6200 |
| hypercenter | C4 | #4176⋊C2 |
| hypocenter | #731 | D8 |
| LCS | G ▷ C4×#731 ▷ C2×#731 ▷ #731 | |
| UCS | C1 ◁ C2 ◁ C4 | |
126 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 24 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=cd,c^2=d,f^3=h,b^a=bc,c^a=cd,f^a=f^2h^2,g^a=g^2,h^a=h^2,e^b=e^2,g^b=g^2h,f^e=fg,g^e=gh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 151);