Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #88935 |
| commutator, cosocle | C3×C3⋊(C3⋊C8) | C22 |
| Frattini, hypercenter | C4 | D32⋊D3 |
| Fitting | C33×C4 | D4 |
| socle | C2×C33 | D8 |
| hypocenter | C33 | D16 |
| LCS | G ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
132 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 19 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|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=f^2g,h^a=h^2,c^b=cd,d^b=de,f^b=f^2g^2,f^c=f^2,g^c=g^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63734, #13165349, #13165355, … |
| maximal normal subgroup of | D3×#63734, #13165349, #13165355, … |
| maximal quotient of | #13154954, #13154955, #13154956, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2732);