Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of #11724050 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊(C22⋊C9) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C2×C3 |
| Frattini | C3 | C3×#1020 |
| Fitting | C24×C32 | D3 |
| socle | C24×C3 | C3×D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 | |
229 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 229 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) = 36 |
Polycyclic presentation:
2,3^3,2^4|c^3=d^2,c^a=c^2d,d^a=d^2,e^a=f,f^a=e,g^a=h,h^a=g,c^b=cd,e^c=f,f^c=ef,g^c=gh,h^c=g46 conjugacy classes — view the full character table.
| maximal subgroup of | (#64188⋊C2)⋊C3, C2×#11724050, #72213207, … |
| maximal normal subgroup of | (#64188⋊C2)⋊C3, C2×#11724050, C5×#11724050, … |
| maximal quotient of | #10593105⋊C3, #10593106⋊C3, #13164526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4017);