Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #67077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C5×C9 ⋊ D4 |
| Sylow | C9 ⋊ D20 |
| Sylow | C5 ⋊ #611 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×D9 |
| commutator, Fitting, cosocle | C2×C5×C9 | C22 |
| Frattini | C2×C3 | D3×D5 |
| socle | C2×C3×C5 | C2×D3 |
| hypocenter | C5×C9 | D4 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,5,3)(7,9)(8,12)(10,11)(13,14)(15,16,17,18);(2,3)(4,5)(6,11,13,12,7,9,8,14,10)(16,18)Polycyclic presentation:
2^3,3,5,3|b^2=c,d^3=f^2,b^a=bc,e^a=e^4,d^b=d^2f,f^b=f^245 conjugacy classes — view the full character table.
| maximal subgroup of | #10623662, #10623663, #10623665, … |
| maximal normal subgroup of | #10623662, #10623663, #10623665, … |
| maximal quotient of | #10623549, #10623551, #10623553, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 13);