Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = D42×AGL(1,𝔽5) (order 1280) · Inn(G) = C2×C22⋊D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C5 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×C22⋊D5 |
| commutator, cosocle | C4×C5 | C23 |
| Frattini | C4 | C22×D5 |
| Fitting, prime radical | C5×#41 | C2 |
| socle | C2×C5 | C2×D4 |
| hypercenter | #41 | D5 |
| hypocenter | C5 | #137 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #41 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,4,5,3)(7,12)(8,15)(9,17)(11,18)(13,14)(19,20);(6,8)(7,11)(9,16)(10,15)(12,20)(13,18)(14,19)(17,21);(2,3)(4,5)(6,7,10,14)(8,11,15,19)(9,12,17,13)(16,20,21,18)Polycyclic presentation:
2^5,5|a^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^4,c^b=ce,d^c=de31 conjugacy classes — view the full character table.
| maximal subgroup of | #65114, #65115, #65430, … |
| maximal normal subgroup of | #65114, #65115, #65430, … |
| maximal quotient of | #65284, #65285, #65293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 172);