Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | D3×C3⋊D3 |
| commutator, socle | C2×C33 | C22×C4 |
| Frattini | C4 | C2×D3×C3⋊D3 |
| Fitting | C33×C8 | C22 |
| prime radical | #89020 | C2 |
| cosocle | C33×C4 | C23 |
| hypocenter | C33 | #131 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C8 | |
460 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 102 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|c^2=d,d^2=e,b^a=be,h^a=h^2,f^b=f^2,g^b=g^2132 conjugacy classes — view the full character table.
| maximal subgroup of | #13155191, #13155192, #13158747, … |
| maximal normal subgroup of | #13155191, #13155192, #13158747, … |
| maximal quotient of | #13159559, #13159561, #13159563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 3298);