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) ⋊ #103 |
123 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 123 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3^3|a^2=e,b^2=d,c^2=e,d^2=e,b^a=bc,c^a=ce,d^a=de,f^a=f^2,g^a=gh,h^a=h^2,f^b=gh^2,g^b=fh^2,h^b=h^2,f^c=f^2,g^c=g^2h^2,g^f=gh37 conjugacy classes — view the full character table.
| maximal subgroup of | #13146468, #13146470, #13146472, … |
| maximal normal subgroup of | #13146468, #13146470, #13146472, … |
| maximal quotient of | #13136668, #13136681, #13136684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 663);