Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #67150 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C32×C5 ⋊ Q8 |
| Sylow | C32 ⋊ Dic10 |
| Sylow | C5 ⋊ #627 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D15 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C32×C5 | C22 |
| hypocenter | C32×C5 | Q8 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
54 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,5,3)(6,8,9,13)(7,10,11,12)(15,16)(17,18,19);(2,3)(4,5)(6,7,9,11)(8,12,13,10)(14,15,16)(18,19)Polycyclic presentation:
2^3,3^2,5|a^2=c,b^2=c,b^a=bc,e^a=e^2,f^a=f^4,d^b=d^251 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#627, D3×#1204, #10624038, … |
| maximal normal subgroup of | D5×#627, D3×#1204, #10624038, … |
| maximal quotient of | #10623814, #10623815, Dic3⋊Dic15, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 83);