Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) = C4×#64241 (order 1152) · Inn(G) = D32 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C2×C5×D32 |
| unambiguous semidirect product | (C3×D3)⋊(C22×C5) |
| unambiguous semidirect product | (C3⋊D3)⋊(C22×C5) |
| Fitting, socle | C2×C32×C5 ⋊ C22 |
| derived, Sylow | C32 ⋊ C23×C5 |
| Hall | C32×C5 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D32 |
| commutator, cosocle, hypocenter | C32 | C23×C5 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C5 | C22 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C5 | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(4,5)(6,7,8)(9,10,11,12,13);(3,4,5)(7,8);(1,2)90 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D32⋊C4, D32⋊Dic5, D32×Dic5, … |
| maximal normal subgroup of | C5×D32⋊C4, D32⋊Dic5, D32×Dic5, … |
| maximal quotient of | C5×D3×Dic6, C5×#3795, C5×#3796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 153);