Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #67138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ Q8 |
| Sylow | C32 ⋊ Dic10 |
| Sylow | C5 ⋊ #634 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C3⋊D3 |
| commutator, Fitting, socle, prime radical, cosocle | C2×C32×C5 | C22 |
| hypocenter | C32×C5 | Q8 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,4,5,3)(6,8,9,13)(7,10,11,12)(15,16)(18,19);(2,3)(4,5)(6,7,9,11)(8,12,13,10)(17,18,19);(14,15,16)Polycyclic presentation:
2^3,3^2,5|a^2=c,b^2=c,b^a=bc,f^a=f^4,d^b=d^2,e^b=e^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1204, #10624038, #10624050, … |
| maximal normal subgroup of | D3×#1204, #10624038, #10624050, … |
| maximal quotient of | #10623773, #10623774, #10623775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 71);