Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#61451 (order 1536) · Inn(G) = #809 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×#108 ⋊ 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)(4,5,6,11,9,12,8,17)(7,15,13,10,16,18,14,19);(2,3)(5,10)(6,8)(7,14)(11,15)(12,19)(13,16)(17,18)Polycyclic presentation:
2^6,3|b^2=c,c^2=f,d^2=ef,e^2=f,b^a=bd,c^a=cf,d^a=de,e^a=ef,g^a=g^2,d^b=def,e^b=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69242, #69244, D3×#361, … |
| maximal normal subgroup of | #69242, #69244, D3×#361, … |
| maximal quotient of | #67465, #67473, #67476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 54);