Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | #737 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10619907 |
| commutator, Fitting, cosocle | C4×#737 | C22 |
| Frattini | C3×C4 | #1113 |
| socle | C2×C3 | #6295 |
| hypercenter | C4 | #4207⋊C2 |
| hypocenter | #737 | D8 |
| LCS | G ▷ C4×#737 ▷ C2×#737 ▷ #737 | |
| UCS | C1 ◁ C2 ◁ C4 | |
171 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 34 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|c^2=d,e^3=h,b^a=bc,c^a=cd,e^a=e^2h^2,g^a=g^2,h^a=h^2,c^b=cd,e^b=e^2h^2,f^b=f^2,h^b=h^2,g^f=gh^269 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1474);