Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽5) (order 640) · Inn(G) = D4×D5 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×Q16 ⋊ C2 |
| Sylow | C5 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D5 |
| commutator, cosocle | C4×C5 | C23 |
| Frattini | C4 | C22×D5 |
| Fitting | C5×Q16 | C2 |
| socle | C2×C5 | C2×D4 |
| prime radical | #713 | C2 |
| hypercenter | Q16 | D5 |
| hypocenter | C5 | #137 |
| LCS | G ▷ C4×C5 ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
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)(6,8,10,14)(7,11,13,20)(9,18,19,16)(12,21,17,15);(6,7,10,13)(8,15,14,21)(9,17,19,12)(11,18,20,16);(2,3)(4,5)(8,14)(11,20)(15,21)(16,18)Polycyclic presentation:
2^5,5|b^2=e,c^2=e,d^2=e,c^a=ce,f^a=f^4,c^b=cd,d^b=de,d^c=de25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4064, #66100, #66102, … |
| maximal normal subgroup of | C2×#4064, #66100, #66102, … |
| maximal quotient of | #65078, #65080, #65084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 139);