Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C24×C4×D3 (order 384) · Inn(G) = C2×D3 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D3)⋊C16 |
| Fitting | C2×C3×C16 ⋊ C2 |
| Sylow | C3 ⋊ C22⋊C16 |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(5,7)(8,9,10,13,11,14,16,19,12,15,17,20,18,21,22,23);(2,3)(4,5)(6,7)Polycyclic presentation:
2^6,3|b^2=c,c^2=e,e^2=f,b^a=bd,g^a=g^272 conjugacy classes — view the full character table.
| maximal subgroup of | #69052, #69053, C16×D12, … |
| maximal normal subgroup of | #69052, #69053, C16×D12, … |
| maximal quotient of | #67503, #67506, #67522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 66);