Finite Group DatabasebetaFinite Group Databasebeta| order | 80 |
| isoclinism | isoclinic class of Dic10 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽5) (order 320) · Inn(G) = C2×D5 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×C5 ⋊ C2 |
| Sylow | C5 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D5 |
| commutator, socle, cosocle | C2×C5 | C23 |
| Frattini | C2 | C22×D5 |
| Fitting | C2×C4×C5 | C2 |
| hypercenter | C2×C4 | D5 |
| hypocenter | C5 | #41 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
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)(7,11)(10,13);(6,8,9,12)(7,10,11,13);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)Polycyclic presentation:
2^4,5|b^2=d,c^a=cd,e^a=e^426 conjugacy classes — view the full character table.
| maximal subgroup of | #3937, #3957, #4047, … |
| maximal normal subgroup of | #3937, #3957, #4047, … |
| maximal quotient of | C4×Dic10, #4016, #4018, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(80, 38);