Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of D80 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#442 (order 1280) · Inn(G) = D40 · Out(G) = C42 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C5×C16 ⋊ C2 |
| Sylow | C5 ⋊ SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D40 |
| commutator, cosocle | C5×C8 | C22 |
| Frattini | C8 | C2×D5 |
| Fitting | C5×C16 | C2 |
| socle | C2×C5 | D8 |
| prime radical | D40 | C2 |
| hypercenter | C16 | D5 |
| hypocenter | C5 | SD32 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,4,5,3)(6,7,12,18,17,21,8,11,10,14,15,20,9,13,16,19);(2,3)(4,5)(7,11)(8,15)(9,17)(12,16)(13,20)(14,19)(18,21)Polycyclic presentation:
2^5,5|b^2=cd,c^2=d,d^2=e,b^a=bc,c^a=cde,d^a=de,f^a=f^443 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#3932, #65193, #65197, … |
| maximal normal subgroup of | C2×#3932, #65193, #65197, … |
| maximal quotient of | #64723, #64725, #64729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 7);