Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C52⋊C3 ⋊ D4 |
| Sylow | C52 ⋊ #625 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×#3874 |
| commutator, cosocle | C2×C3×C52⋊C3 | C22 |
| Fitting, socle | C2×C3×C52 | C2×D3 |
| hypocenter | C3×C52⋊C3 | D4 |
| LCS | G ▷ C2×C3×C52⋊C3 ▷ C3×C52⋊C3 | |
| UCS | C1 ◁ C2 | |
102 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 102 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^3,3^2,5^2|b^a=bc,d^a=d^2,f^a=fg^2,g^a=g^4,e^b=e^2,f^d=f^3g^3,g^d=f^4g63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 343);