Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ C2×D8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×#157⋊D3 |
| commutator, cosocle | C3×C3⋊Dic3 | C23 |
| Fitting | C33×C4 | D4 |
| socle | C2×C33 | C2×D4 |
| hypercenter | C4 | #157⋊D3 |
| hypocenter | C33 | C2×D8 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
292 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^5,3^3|b^2=de,c^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^2,g^a=f^2g,h^a=h^2,f^b=f^2g^2,g^b=f^2g,f^d=f^2,g^d=g^248 conjugacy classes — view the full character table.
| maximal subgroup of | #13165113, #13165131, #13165194, … |
| maximal normal subgroup of | #13165113, #13165131, #13165194, … |
| maximal quotient of | #13167351, #13167352, #13167353, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4303);