Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #88668 |
| commutator, cosocle | C3×C4×C9 | C23 |
| Frattini | C3×C4 | C2×D32 |
| Fitting | C3×D4×C9 | C22 |
| socle | C2×C32 | D3×D4 |
| hypercenter | D4 | D3×D9 |
| hypocenter | C3×C9 | #135 |
| LCS | G ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
226 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 49 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:
2^5,3^3|b^2=e,d^2=e,g^3=h,c^a=cd,d^a=de,g^a=g^2h^2,h^a=h^2,c^b=cd,d^b=de,f^b=f^2,d^c=de69 conjugacy classes — view the full character table.
| maximal subgroup of | D9×#912, #13149900, #13149905, … |
| maximal normal subgroup of | D9×#912, #13149900, #13149905, … |
| maximal quotient of | #13137557, #13137559, #13137577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1153);