Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#3109 (order 768) · Inn(G) = #821 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×M32 ⋊ C2 |
| Sylow | C3 ⋊ #360 |
58 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)(4,6,14,5,9,16,7,11,10,17,13,12,8,19,18,15);(2,3)(4,5)(6,13)(7,16)(8,11)(9,15)(10,12)(14,17)(18,19)Polycyclic presentation:
2^6,3|b^2=cd,c^2=ef,d^2=f,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 | #69090, #69091, #69120, … |
| maximal normal subgroup of | #69090, #69091, #69120, … |
| maximal quotient of | #67516, #67517, #67526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 77);