Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5)×S4 (order 960) · Inn(G) = C22×D5 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C5×Q8 ⋊ C2 |
| Sylow | C5 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D5 |
| commutator, socle, cosocle | C2×C5 | C23 |
| Fitting | C5×Q8 | C2 |
| hypercenter | Q8 | D5 |
| hypocenter | C5 | #41 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
40 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)(6,8,9,12)(7,10,11,13);(6,7,9,11)(8,13,12,10);(2,3)(4,5)(8,12)(10,13)Polycyclic presentation:
2^4,5|b^2=d,c^2=d,c^a=cd,e^a=e^4,c^b=cd20 conjugacy classes — view the full character table.
| maximal subgroup of | #4010, #4060, #4062, … |
| maximal normal subgroup of | #4010, #4060, #4062, … |
| maximal quotient of | #4036, #4038, #4039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 42);