Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of D48 |
| automorphism | Aut(G) = C22×D3×#442 (order 1536) · Inn(G) = D24 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×C16 ⋊ C2 |
| Sylow | C3 ⋊ #357 |
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)(4,5,6,7)(8,9,14,20,11,15,18,21,12,16,22,23,17,19,10,13);(2,3)(9,13)(10,14)(11,17)(15,23)(16,21)(18,22)(19,20)Polycyclic presentation:
2^6,3|b^2=cde,c^2=ef,e^2=f,b^a=bc,c^a=ce,e^a=ef,g^a=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D48, #69058, #69059, … |
| maximal normal subgroup of | C4×D48, #69058, #69059, … |
| maximal quotient of | #67504, #67526, #67560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 68);