Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C92 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10619530 |
| commutator, Fitting, cosocle | C4×C92 | C22 |
| Frattini | C32×C4 | D32 |
| socle | C2×C32 | #626 |
| prime radical | C4⋊(C9⋊D9) | C2 |
| hypercenter | C4 | D92 |
| hypocenter | C92 | SD16 |
| LCS | G ▷ C4×C92 ▷ C2×C92 ▷ C92 | |
| UCS | C1 ◁ C2 ◁ C4 | |
146 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3^4|a^2=d,b^2=c,c^2=d,e^3=h,f^3=g,b^a=bc,c^a=cd,f^a=f^2g^2,g^a=g^2,e^b=e^2h^2,h^b=h^2123 conjugacy classes — view the full character table.
| maximal subgroup of | #72219870, #72219899, #72219913, … |
| maximal normal subgroup of | #72219870, #72219899, #72219913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 86);