Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #106 |
106 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=d,b^2=de,c^2=e,b^a=bc,c^a=ce,f^a=f^2,g^a=gh,h^a=h^2,c^b=ce,f^b=gh^2,g^b=fh^2,h^b=h^2,f^c=f^2,g^c=g^2h^2,g^f=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #13146179, #13146507, #13146508, … |
| maximal normal subgroup of | #13146179, #13146507, #13146508, … |
| maximal quotient of | #13136670, #13136693, #13136719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 670);