Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4466 |
| automorphism | Aut(G) = D3×#511819 (order 3072) · Inn(G) = C4×Dic3 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C3⋊#338 |
| Sylow | C3 ⋊ #338 |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,6,7,16,9,17,8,15)(5,10,12,14,13,18,11,19);(2,3)(4,5,7,11,9,13,8,12)(6,14,15,18,17,19,16,10)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,c^2=f,d^2=f,e^2=f,b^a=be,d^a=df,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #67535, #67537, #67779, … |
| maximal normal subgroup of | #67535, #67537, #67779, … |
| maximal quotient of | #67660, #67670, #67679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 119);