Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C5 |
No aliases for this group.
| Hall | C32×C52 ⋊ D4 |
| Sylow | C52 ⋊ #626 |
| Sylow | C32 ⋊ C4⋊(C5⋊D5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊(C5⋊D5) |
| commutator, Fitting, socle, cosocle | C2×C32×C52 | C22 |
| hypocenter | C32×C52 | D4 |
| LCS | G ▷ C2×C32×C52 ▷ C32×C52 | |
| UCS | C1 ◁ C2 | |
296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 300 |
Polycyclic presentation:
2^3,3^2,5^2|b^2=c,b^a=bc,d^a=d^2,f^a=f^4,g^a=g^4,e^b=e^2231 conjugacy classes — view the full character table.
| maximal subgroup of | C5⋊D5×#626, #72231387, #72231388, … |
| maximal normal subgroup of | C5⋊D5×#626, #72231387, #72231388, … |
| maximal quotient of | C2×#14268293 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 487);