Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #137 |
168 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 26 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|b^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^2,g^a=gh,h^a=h^2,d^b=de,f^b=gh^2,g^b=fh^2,h^b=h^2,f^d=f^2,g^d=g^2h^2,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #13146610, #13146700, #13164308, … |
| maximal normal subgroup of | #13146610, #13146700, #13164308, … |
| maximal quotient of | #13146508, #13146511, #13146518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2274);