Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#61451 (order 1536) · Inn(G) = #821 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×M32 ⋊ C2 |
| Sylow | C3 ⋊ #361 |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,13,14,16,19,17,8,10,15,9,7,11,12,18);(2,3)(5,9)(6,12)(7,14)(10,13)(11,18)(15,19)(16,17)Polycyclic presentation:
2^6,3|b^2=cd,c^2=ef,e^2=f,b^a=bc,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,d^b=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #69091, #69116, #69126, … |
| maximal normal subgroup of | #69091, #69116, #69126, … |
| maximal quotient of | #67516, #67527, #67560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 75);