Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#87524 (order 7680) · Inn(G) = C22×D5 · Out(G) = #5912 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C24×C5 ⋊ C2 |
| Sylow | C5 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D5 |
| commutator, socle, cosocle | C22×C5 | C23 |
| Fitting | C24×C5 | C2 |
| hypercenter | C24 | D5 |
| hypocenter | C5 | #120 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C24 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2,4,5,3)(7,9);(11,13);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)Polycyclic presentation:
2^5,5|b^a=be,c^a=cd,f^a=f^446 conjugacy classes — view the full character table.
| maximal subgroup of | #64934, #65321, #65824, … |
| maximal normal subgroup of | #64934, #65321, #65824, … |
| maximal quotient of | #65499, #65502, #65506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 174);