Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| 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) = C22×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic10 |
| unambiguous semidirect product | C5⋊(C3×Q8) |
| Hall | C3×C5 ⋊ Q8 |
| Sylow | C5 ⋊ C3×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D5 |
| commutator, cosocle | C2×C5 | C22×C3 |
| Frattini | C2 | C2×C3×D5 |
| Fitting | C3×C4×C5 | C2 |
| socle, prime radical | C2×C3×C5 | C22 |
| hypercenter | C3×C4 | D5 |
| hypocenter | C5 | C3×Q8 |
| LCS | G ▷ C2×C5 ▷ C5 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3)(4,5,6,7)(8,11,10,9)(12,13,14,15,16);(4,8,6,10)(5,9,7,11)(13,16)(14,15)39 conjugacy classes — view the full character table.
| maximal subgroup of | #6656, #6658, #6661, … |
| maximal normal subgroup of | #6656, #6658, #6661, … |
| maximal quotient of | C3×#683, C3×C4⋊Dic5, C2×C3×Dic10, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 16);