Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C24×AGL(1,𝔽5) (order 320) · Inn(G) = C2×D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C5 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D5 |
| commutator | C2×C5 | C2×C4 |
| Fitting | C2×C4×C5 | C2 |
| socle, prime radical, cosocle | C22×C5 | C22 |
| hypercenter | C2×C4 | D5 |
| hypocenter | C5 | C4⋊C4 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,4,5,3)(6,7,8,9)(11,13);(2,3)(4,5)(10,11,12,13)Polycyclic presentation:
2^4,5|a^2=d,b^2=c,b^a=bd,e^a=e^426 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic10, #4016, #4018, … |
| maximal normal subgroup of | C4×Dic10, #4016, #4018, … |
| maximal quotient of | #3939, #3940, #3946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 12);