Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10619751 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| derived | #66357 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #10619751 |
| commutator, hypocenter | #66357 | C4 |
| Frattini | C2×C32 | C3⋊S4 |
| Fitting | C23×C3×C9 | D3 |
| socle | C23×C3 | #272 |
| prime radical | C2×C22⋊H(3,𝔽3) | D3 |
| cosocle | C2×#66357 | C2 |
| LCS | G ▷ #66357 | |
| UCS | C1 ◁ C2 | |
134 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 25 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:
2^2,3^4,2^2|a^2=b,d^3=f,c^a=c^2,d^a=d^2f^2,e^a=ef^2,f^a=f^2,g^a=gh,d^c=de,e^c=ef,g^c=h,h^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 728);