Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C33 ⋊ D16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #88790 |
| commutator, Fitting, cosocle | C33×C8 | C22 |
| Frattini, hypercenter | C8 | D3×C3⋊D3 |
| socle | C2×C33 | D8 |
| hypocenter | C33 | D16 |
| LCS | G ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
256 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 58 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=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,h^a=h^2,f^b=f^2,g^b=g^287 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63546, #13155197, #13155220, … |
| maximal normal subgroup of | D3×#63546, #13155197, #13155220, … |
| maximal quotient of | #13145491, #13145495, #13145496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1972);