Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×AGL(1,ℤ/8) (order 768) · Inn(G) = D3×D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#108 ⋊ C2 |
| Sylow | C3 ⋊ #443 |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,8,17,9,15,18,10)(5,11,12,19,14,6,13,16);(4,6,8,16,9,11,18,19)(5,10,13,15,14,17,12,7);(2,3)(4,5)(6,15)(7,11)(8,12)(9,14)(10,16)(13,18)(17,19)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^2,c^b=cef,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #69238, #69239, #69262, … |
| maximal normal subgroup of | #69238, #69239, #69262, … |
| maximal quotient of | #68516, #68521, #68522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 454);