Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#63011 (order 1536) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#117 ⋊ C2 |
| Sylow | C3 ⋊ #410 |
158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,7)(5,10,11,14)(6,12)(8,15)(9,16,17,18)(13,19);(4,6,8,13)(5,9,11,17)(7,12,15,19)(10,16,14,18);(2,3)(4,5)(6,9)(7,14)(8,11)(10,15)(12,18)(13,17)(16,19)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,c^a=ce,d^a=df,g^a=g^2,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #68698, #68704, #68707, … |
| maximal normal subgroup of | #68698, #68704, #68707, … |
| maximal quotient of | #69431, #69432, #69433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 565);