Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C22 |
| Frattini | C2×C3 | C2×C3⋊S4 |
| Fitting | C2×C32×Q8 | D3 |
| socle | C2×C32 | C2×S4 |
| hypercenter | C22 | C3⋊((C22⋊C9)⋊C2) |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
208 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^2,3^3,2^3|c^3=e,f^2=h,g^2=h,b^a=bh,c^a=c^2e^2,d^a=d^2,e^a=e^2,f^a=gh,g^a=fh,f^c=g,g^c=fgh,g^f=gh69 conjugacy classes — view the full character table.
| maximal subgroup of | #13164184, #13164188, D3×#63689, … |
| maximal normal subgroup of | #13164184, #13164188, D3×#63689, … |
| maximal quotient of | #13146992, #13146993, #13146998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2352);