Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of #700 |
| automorphism | Aut(G) = D4×AGL(1,𝔽5) (order 160) · Inn(G) = C2×AGL(1,𝔽5) · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72221731 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C5 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽5) |
| commutator, socle | C2×C5 | C2×C4 |
| Fitting | C4×C5 | C4 |
| prime radical | C2×D5 | C4 |
| cosocle | Dic5 | C22 |
| hypercenter | C4 | AGL(1,𝔽5) |
| hypocenter | C5 | M16 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3,5,4)(6,8,10,9)(7,11,12,13);(2,3,4,5)(6,7,9,11,10,12,8,13)Polycyclic presentation:
2^4,5|a^2=c,b^2=d,c^2=d,b^a=bd,e^a=e^2,e^c=e^414 conjugacy classes — view the full character table.
| maximal subgroup of | #3995, #3996, #4008, … |
| maximal normal subgroup of | #3995, #3996, #4008, … |
| maximal quotient of | C4⋊#170, #4002, #4003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 29);