Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#909 |
| Fitting | C2×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#137 |
258 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(3,4,5)(6,9,10,17,11,18,8,15)(7,13,12,16,14,21,19,20);(1,2)(6,8,11,10)(7,12,14,19)(9,17,18,15)(13,20,21,16);(4,5)(6,7)(8,12)(9,16)(10,19)(11,14)(13,17)(15,21)(18,20);(1,2)42 conjugacy classes — view the full character table.
| maximal subgroup of | #68707, #68725, #68762, … |
| maximal normal subgroup of | #68707, #68725, #68762, … |
| maximal quotient of | C2×#4702, C2×#4710, C2×#4714, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1306);