Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #67077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C5×C9 ⋊ Q8 |
| Sylow | C9 ⋊ Dic10 |
| Sylow | C5 ⋊ Dic18 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D5×D9 |
| commutator, Fitting, cosocle | C2×C5×C9 | C22 |
| Frattini | C2×C3 | D3×D5 |
| socle, prime radical | C2×C3×C5 | C2×D3 |
| hypocenter | C5×C9 | Q8 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2,4,5,3)(7,9)(8,12)(10,11)(13,14)(15,17,18,22)(16,19,20,21);(2,3)(4,5)(6,11,13,12,7,9,8,14,10)(15,16,18,20)(17,21,22,19)Polycyclic presentation:
2^3,3,5,3|a^2=c,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 | D9×Dic10, D45⋊Q8, D5×Dic18, … |
| maximal normal subgroup of | D9×Dic10, D45⋊Q8, D5×Dic18, … |
| maximal quotient of | #10623557, Dic9⋊Dic5, #10623559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 7);