Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C4×#72209832 (order 12288) · Inn(G) = C22 · Out(G) = C4×#87588⋊C2 |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×D4 |
| unambiguous semidirect product | C4⋊(C23×C5) |
| unambiguous semidirect product | C22⋊(C23×C5) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(9,10,11,12,13);(3,4);(5,6,7,8);(6,8)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 229);