Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #10619744⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #10619744⋊C2 |
| unambiguous semidirect product | #10619756⋊C2 |
| unambiguous semidirect product | #10619772⋊C2 |
| derived | #66361 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #66361 | C22 |
| Frattini | C32 | D3×S4 |
| Fitting | C22×H(3,𝔽3) | C2×D3 |
| socle | C22×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ #66361 | |
| UCS | C1 | |
207 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2^2,3^4,2^2|c^3=f^2,d^a=d^2,e^a=e^2f,c^b=c^2f,e^b=e^2,f^b=f^2,g^b=gh,d^c=de,g^c=h,h^c=gh,e^d=ef28 conjugacy classes — view the full character table.
| maximal subgroup of | #72276237 |
| maximal normal subgroup of | #72276237 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1787);