Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ #107 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | #157⋊D3 |
| commutator | C3×C3⋊Dic3 | C2×C4 |
| Fitting, socle, prime radical | C22×C33 | D4 |
| cosocle | C2×C3×C3⋊Dic3 | C22 |
| hypocenter | C33 | #107 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^5,3^3|a^2=d,b^2=ce,c^2=e,b^a=bc,c^a=ce,f^a=f^2,g^a=f^2g,h^a=h^2,f^b=f^2g^2,g^b=f^2g,f^c=f^2,g^c=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#63743, #13165482, #13165483, … |
| maximal normal subgroup of | D3×#63743, #13165482, #13165483, … |
| maximal quotient of | #13155086, #13155112, #13155113, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2764);