Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = #67 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #416 |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7)(9,14)(11,19)(13,20)(15,16)(17,23)(21,22);(8,10,12,18)(9,13,16,22)(11,17,19,23)(14,20,15,21);(2,3)(8,9)(10,13)(11,14)(12,16)(15,19)(17,20)(18,22)(21,23)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,e^2=f,c^a=ce,e^a=ef,g^a=g^2,e^c=ef48 conjugacy classes — view the full character table.
| maximal subgroup of | #68698, #68700, #68739, … |
| maximal normal subgroup of | #68698, #68700, #68739, … |
| maximal quotient of | #69431, #69443, C4×#809, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 561);