Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D5×C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| derived, Hall | C32×C5 ⋊ C2×C4 |
| Sylow | C32 ⋊ C4×D5 |
| Sylow | C5 ⋊ C4×C3⋊D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C3⋊D3 |
| commutator, hypocenter | C32×C5 | C2×C4 |
| Fitting, socle, cosocle | C2×C32×C5 | C22 |
| prime radical | C2×C3⋊D15 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,5,3)(6,8,9,7)(11,12)(14,15);(2,3)(4,5)(6,7,9,8)(13,14,15);(10,11,12)Polycyclic presentation:
2^3,3^2,5|a^2=c,b^2=c,f^a=f^4,d^b=d^2,e^b=e^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10623967, #10623969, #10623981, … |
| maximal normal subgroup of | #10623967, #10623969, #10623981, … |
| maximal quotient of | #10623753, #10623756, Dic5×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 67);