Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C2×C5×C3⋊Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C5×Dic3) |
| derived, Sylow | C32 ⋊ C2×C4×C5 |
| Hall | C32×C5 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C5 | C3⋊D3 |
| commutator, hypocenter | C32 | C2×C4×C5 |
| Frattini | C2 | C2×C5×C3⋊D3 |
| Fitting, socle, prime radical | C22×C32×C5 | C2 |
| cosocle | C2×C32 | C22×C5 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C5 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 360 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2)(4,5)(7,8)(9,10,11,12)(13,14,15,16,17);(1,2)(3,4,5);(6,7,8)120 conjugacy classes — view the full character table.
| maximal subgroup of | Dic5×C3⋊Dic3, #10623770, #10623772, … |
| maximal normal subgroup of | Dic5×C3⋊Dic3, #10623770, #10623772, … |
| maximal quotient of | C2×C5×C3⋊(C3⋊C8), C5×#3748, C4×C5×C3⋊Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 108);