Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #13049779 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, socle, Sylow | C34 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator | C3⋊(C3⋊(C3⋊D3)) | C2×C4 |
| Fitting, socle, hypocenter | C34 | C22⋊C4 |
| prime radical | #10620215 | C2 |
| cosocle | C3⋊D32 | C22 |
| LCS | G ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34 | |
| UCS | C1 | |
295 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^4,3^4|a^2=d,b^a=bcd,e^a=f^2,f^a=e,g^a=h,h^a=g,e^b=e^2,g^b=g^2,g^c=g^2,h^c=h^2,e^d=e^2,f^d=f^233 conjugacy classes — view the full character table.
| maximal subgroup of | #72213220, #72220438 |
| maximal normal subgroup of | #72213220, #72220438 |
| maximal quotient of | #72259932 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3497);