Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×C3⋊D3 |
| commutator, socle, prime radical, cosocle | C22×C33 | C23 |
| Fitting | C2×C33×C4 | C22 |
| hypercenter | C2×C4 | D3×C3⋊D3 |
| hypocenter | C33 | C4⋊Q8 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
512 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^5,3^3|a^2=d,b^2=d,c^2=e,b^a=bd,c^a=ce,h^a=h^2,f^b=f^2,g^b=g^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #13154786, #13154787, #13156272, … |
| maximal normal subgroup of | #13154786, #13154787, #13156272, … |
| maximal quotient of | #13159830, #13159832, #13159867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3502);