Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C9×#41 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9⋊A4 |
| commutator, cosocle, hypocenter | Q8×C9 | C22×C3 |
| Frattini, socle | C2×C3 | C2×D3×A4 |
| Fitting | C9×#41 | C2×C3 |
| hypercenter | C4 | D9⋊A4 |
| LCS | G ▷ Q8×C9 | |
| UCS | C1 ◁ C2 ◁ C4 | |
143 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 143 conjugacy classes of subgroups — too many to draw. Only its 28 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:
2^2,3,2^2,3,2,3|b^2=g,d^2=g,e^2=g,f^3=h,b^a=bg,f^a=f^2h^2,h^a=h^2,d^c=e,e^c=de,f^c=fh^2,e^d=eg43 conjugacy classes — view the full character table.
| maximal subgroup of | #13147143, #13147145, #13147189, … |
| maximal normal subgroup of | #13147143, #13147145, #13147189, … |
| maximal quotient of | #13147103, C4⋊(D9⋊SL(2,𝔽3)), (D9⋊SL(2,𝔽3))⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2381);