Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #939 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#137 ⋊ C2 |
| Sylow | C3 ⋊ #471 |
104 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,9,16)(5,11,12,19)(7,15,8,13)(10,18,17,14);(4,5)(6,14)(7,10)(8,17)(9,12)(11,15)(13,19)(16,18);(2,3)(5,10)(6,13)(7,8)(12,17)(14,18)(15,16)Polycyclic presentation:
2^6,3|c^2=f,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,d^c=df,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | #80294, #80296, #80301, … |
| maximal normal subgroup of | #80294, #80296, #80301, … |
| maximal quotient of | #70299, #70302, #70303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 762);