Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of #3958 |
| automorphism | Aut(G) = AGL(1,𝔽5)×#442 (order 1280) · Inn(G) = #686 · Out(G) = C42 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #686 |
| commutator, cosocle | C5×C8 | C22 |
| Frattini | C8 | C2×D5 |
| Fitting | C5×Q16 | C2 |
| socle, prime radical | C2×C5 | D8 |
| hypercenter | Q16 | D5 |
| hypocenter | C5 | Q32 |
| LCS | G ▷ C5×C8 ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,4,5,3)(6,8,11,19)(7,12,15,29)(9,21,24,37)(10,26,22,18)(13,31,32,36)(14,33,25,28)(16,34,35,20)(17,30,27,23);(2,3)(4,5)(6,7,11,15)(8,16,19,35)(9,20,24,34)(10,25,22,14)(12,27,29,17)(13,30,32,23)(18,36,26,31)(21,33,37,28)Polycyclic presentation:
2^5,5|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^4,c^b=cd,d^b=de25 conjugacy classes — view the full character table.
| maximal subgroup of | #65204, #65205, D5×Q32, … |
| maximal normal subgroup of | #65204, #65205, D5×Q32, … |
| maximal quotient of | #64709, #64712, #64784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 36);