Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25×C3 |
No aliases for this group.
| Fitting | #41×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini, socle | C2×C3 | C23×C3×D3 |
| Fitting | #41×H(3,𝔽3) | C2 |
| hypercenter | #41 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 | |
621 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^4,3,2,3^2|d^2=f,d^a=df,g^a=g^2,h^a=h^2,c^b=cf,g^e=gh91 conjugacy classes — view the full character table.
| maximal subgroup of | #13141068, #13141070, #13141077, … |
| maximal normal subgroup of | #13141068, #13141070, #13141077, … |
| maximal quotient of | #10596003⋊C3, #10595990⋊C3, #10595992⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2605);