Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10620047⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10620047⋊C2 |
| unambiguous semidirect product | #10620072⋊C2 |
| derived | #66437 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #66437 | C22 |
| Frattini | C3 | #89101 |
| Fitting | C22×H(3,𝔽3) | C2×D3 |
| socle | C22×C3 | #1113 |
| LCS | G ▷ #66437 | |
| UCS | C1 | |
241 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2^2,3^4,2^2|c^3=f,c^a=c^2f^2,d^a=d^2,f^a=f^2,g^a=gh,d^b=d^2,e^b=e^2,g^c=h,h^c=gh,e^d=ef^232 conjugacy classes — view the full character table.
| maximal subgroup of | #72276229 |
| maximal normal subgroup of | #72276229 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2937);