Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #89101 |
| commutator, cosocle | C2×C32×A4 | C22 |
| Fitting, socle | C23×C32 | C2×D3 |
| hypocenter | C32×A4 | D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
327 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^3,3^3,2^2|b^a=bc,d^a=d^2,e^a=e^2,g^a=h,h^a=g,e^b=e^2,f^b=f^2,g^d=h,h^d=gh42 conjugacy classes — view the full character table.
| maximal subgroup of | #13180286, #13180287, S4×#626, … |
| maximal normal subgroup of | #13180286, #13180287, S4×#626, … |
| maximal quotient of | #13166078, #13166079, #13166080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4183);