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.
| 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)(5,11)(6,12)(8,16)(9,10)(13,14)(15,19)(17,18);(4,6)(5,10)(7,13)(8,14)(9,15)(11,18)(12,17)(16,19);(2,3)(4,5,8,12,9,13,18,19)(6,10,14,17,15,7,11,16)Polycyclic presentation:
2^6,3|a^2=d,d^2=f,e^2=f,g^a=g^2,c^b=ce,e^b=ef,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #69238, #69239, #69246, … |
| maximal normal subgroup of | #69238, #69239, #69246, … |
| maximal quotient of | #70287, #70289, #70291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 755);