Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | 2−1+4⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×2−1+4⋊C3)⋊C3 |
| Fitting | C32×2−1+4 ⋊ C3 |
| Sylow | 2−1+4 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C22×C3×A4 |
| commutator, cosocle | C3×Q8 | C22×C32 |
| Fitting | C32×2−1+4 | C3 |
| hypercenter | C32×D4 | A4 |
| hypocenter | Q8 | C22×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C32×D4 | |
263 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^2,3^3,2^3|f^2=h,g^2=h,b^a=bh,d^c=de,f^c=g,g^c=fgh,g^f=gh91 conjugacy classes — view the full character table.
| maximal subgroup of | #10593664⋊C3, #10593665⋊C3, #13147081, … |
| maximal normal subgroup of | #10593664⋊C3, #10593665⋊C3, #13147081, … |
| maximal quotient of | #519⋊H(3,𝔽3), #520⋊H(3,𝔽3), #548⋊H(3,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2547);