Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of Dic90 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C5×C9 ⋊ C2 |
| Hall | C5×C9 ⋊ D4 |
| Sylow | C9 ⋊ C22⋊D5 |
| Sylow | C5 ⋊ #611 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D45 |
| commutator, cosocle | C2×C5×C9 | C22 |
| Frattini | C2×C3 | C2×D15 |
| Fitting | C22×C5×C9 | C2 |
| socle | C2×C3×C5 | C2×D3 |
| hypercenter | C22 | D45 |
| hypocenter | C5×C9 | D4 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C2 ◁ C22 | |
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)(6,11,13,12,7,9,8,14,10)(16,18);(2,3)(4,5)(7,9)(8,12)(10,11)(13,14)(15,16)(17,18)Polycyclic presentation:
2^3,3,5,3|d^3=f^2,b^a=bc,d^a=d^2f,e^a=e^4,f^a=f^293 conjugacy classes — view the full character table.
| maximal subgroup of | #10623672, #10623676, D5×#611, … |
| maximal normal subgroup of | #10623672, #10623676, D5×#611, … |
| maximal quotient of | #10623604, #10623606, #10623607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 29);