Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D3×C3⋊D3 |
| commutator | C2×C33 | C42 |
| Fitting, cosocle | C2×C33×C4 | C22 |
| socle, prime radical | C22×C33 | C2×C4 |
| hypocenter | C33 | #97 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
328 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=c,b^2=d,c^2=e,b^a=be,h^a=h^2,f^b=f^2,g^b=g^2132 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63554, #13155275, Dic3×#3711, … |
| maximal normal subgroup of | D3×#63554, #13155275, Dic3×#3711, … |
| maximal quotient of | #13145503, #13145505, #13145549, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1987);