Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C33 |
| unambiguous semidirect product | C4⋊(C3⋊(C3⋊S4)) |
| Fitting | C22×C32×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊S4) |
| commutator, cosocle | C2×C32×A4 | C22 |
| Fitting | C22×C32×C4 | D3 |
| socle | C23×C32 | C2×D3 |
| hypercenter | C4 | C3⋊(C3⋊S4) |
| hypocenter | C32×A4 | D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
592 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3^3,2^2|b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=f^2,g^a=h,h^a=g,g^d=h,h^d=gh78 conjugacy classes — view the full character table.
| maximal subgroup of | #13165928, #13165929, #13165934, … |
| maximal normal subgroup of | #13165928, #13165929, #13165934, … |
| maximal quotient of | C4⋊(C3⋊(C3⋊#226)), C4⋊(C3⋊(C3⋊GL(2,𝔽3))), #13173045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4432);