Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = D3×C22⋊C4 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#100 ⋊ C2 |
| Sylow | C3 ⋊ #413 |
150 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)(4,6,7,13,8,12,14,18)(5,10,11,17,9,15,16,19);(5,9)(6,12)(7,14)(17,19);(2,3)(4,5)(6,10)(7,11)(8,9)(12,15)(13,17)(14,16)(18,19)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,b^a=bf,g^a=g^2,c^b=ce,d^b=df,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #69136, #69137, #69144, … |
| maximal normal subgroup of | #69136, #69137, #69144, … |
| maximal quotient of | #68100, #68102, #68105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 304);