Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4451 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#97 ⋊ C2 |
| Sylow | C3 ⋊ #329 |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,9,7,8,10,11)(13,16,17,18);(2,3)(5,8)(6,10)(12,13)(14,16)(15,17)(18,19)Polycyclic presentation:
2^6,3|b^2=c,c^2=f,d^2=ef,b^a=bd,c^a=ce,d^a=def,g^a=g^2,d^b=df39 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#329, #68342, #68343, … |
| maximal normal subgroup of | D3×#329, #68342, #68343, … |
| maximal quotient of | #67479, #67480, #67520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 23);