Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C33 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×GO+(4,𝔽2) |
| commutator, cosocle | C2×C3×C3⋊D3 | C23 |
| Fitting, socle | C2×C33 | C2×D4 |
| hypocenter | C33 | #120 |
| LCS | G ▷ C2×C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
496 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^5,3^3|c^2=e,c^a=ce,h^a=h^2,c^b=cd,f^b=f^2,g^b=f^2g,f^c=f^2g^2,f^d=f^2,g^d=g^245 conjugacy classes — view the full character table.
| maximal subgroup of | #13180448, #13180450, #13180458, … |
| maximal normal subgroup of | #13180448, #13180450, #13180458, … |
| maximal quotient of | #13165194, #13165195, #13165198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4080);