Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C4×D3×D4 (order 192) · Inn(G) = C2×D3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×Dic6 |
| unambiguous semidirect product | C3⋊(C5×Q8) |
| Hall | C3×C5 ⋊ Q8 |
| Sylow | C3 ⋊ C5×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C5 |
| Frattini | C2 | C2×C5×D3 |
| Fitting | C3×C4×C5 | C2 |
| socle, prime radical | C2×C3×C5 | C22 |
| hypercenter | C4×C5 | D3 |
| hypocenter | C3 | C5×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C5 ◁ C4×C5 | |
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)(6,7,8,9)(10,13,12,11)(14,15,16);(6,10,8,12)(7,11,9,13)(15,16)45 conjugacy classes — view the full character table.
| maximal subgroup of | #6655, #6660, #6661, … |
| maximal normal subgroup of | #6655, #6660, #6661, … |
| maximal quotient of | C5×Dic3⋊C4, C5×C4⋊Dic3, C2×C5×Dic6, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 21);