Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | C32×C52 ⋊ D4 |
| Sylow | C52 ⋊ #638 |
| Sylow | C32 ⋊ #6038 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #11744719 |
| commutator, Fitting, socle, cosocle | C2×C32×C52 | C22 |
| hypocenter | C32×C52 | D4 |
| LCS | G ▷ C2×C32×C52 ▷ C32×C52 | |
| UCS | C1 ◁ C2 | |
252 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
Polycyclic presentation:
2^3,3^2,5^2|b^a=bc,d^a=d^2,e^a=e^2,f^a=f^4,f^b=f^4,g^b=g^4141 conjugacy classes — view the full character table.
| maximal subgroup of | #72229351, C3⋊D3×#6038, #72231468, … |
| maximal normal subgroup of | #72229351, C3⋊D3×#6038, #72231468, … |
| maximal quotient of | C2×#14268269 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 463);