Finite Group DatabasebetaFinite Group Databasebeta| order | 1800 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C52⋊C3 ⋊ Q8 |
| Sylow | C52 ⋊ #627 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #3874⋊D3 |
| commutator, prime radical, cosocle | C2×C3×C52⋊C3 | C22 |
| Fitting, socle | C2×C3×C52 | C2×D3 |
| hypocenter | C3×C52⋊C3 | Q8 |
| LCS | G ▷ C2×C3×C52⋊C3 ▷ C3×C52⋊C3 | |
| UCS | C1 ◁ C2 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 360 |
Polycyclic presentation:
2^3,3^2,5^2|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=fg,g^a=g^4,d^b=d^2,f^b=f^4g^4,f^d=g,g^d=f^4g^451 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1800, 345);