Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C8×H(3,𝔽3) ⋊ C22 |
| Sylow | H(3,𝔽3) ⋊ AGL(1,ℤ/8) |
260 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 44 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:
2^5,3^3|c^2=d,d^2=e,c^a=ce,g^a=g^2,h^a=h^2,c^b=cd,d^b=de,f^b=f^2,h^b=h^2,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11720848, #13149829, #13149830, … |
| maximal normal subgroup of | C2×#11720848, #13149829, #13149830, … |
| maximal quotient of | #13138818, #13138819, #13138820, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 815);