Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | #737 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#4207⋊C2 |
| commutator, cosocle | C2×#737 | C23 |
| Frattini, socle | C2×C3 | C2×#1113 |
| Fitting | C22×#737 | C22 |
| hypercenter | C22 | #4207⋊C2 |
| hypocenter | #737 | #41 |
| LCS | G ▷ C2×#737 ▷ #737 | |
| UCS | C1 ◁ C2 ◁ C22 | |
286 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 50 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=d,f^3=h,c^a=cd,e^a=e^2,f^a=f^2h^2,h^a=h^2,e^b=e^2,g^b=g^2,g^e=gh^272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2502);