Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #66419⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (#4207⋊C2)⋊C4 |
| Sylow | #737 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #66419⋊C2 |
| commutator | #4208 | C2×C4 |
| Frattini, socle | C2×C3 | #157⋊D3 |
| Fitting | C2×#737 | D4 |
| prime radical | C2×#4207⋊C2 | C2 |
| cosocle | C2×#4208 | C22 |
| hypocenter | #737 | C22⋊C4 |
| LCS | G ▷ #4208 ▷ #737 | |
| UCS | C1 ◁ C2 | |
173 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 173 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^4,3^4|b^2=cd,g^3=h,b^a=bc,e^a=e^2,f^a=fh,g^a=g^2h^2,h^a=h^2,e^b=fh^2,f^b=e^2h,e^c=e^2,f^c=f^2h^2,f^e=fh54 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1992);