Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×D3×SD16 |
| Fitting | C2×C3×SD16 ⋊ C2 |
| Sylow | C3 ⋊ C22×SD16 |
298 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 298 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) = 13 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(3,4,5)(7,10)(8,13)(11,12);(1,2)(6,7,9,12)(8,10,13,11);(4,5);(1,2)42 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×SD16, #79649, SD16×D12, … |
| maximal normal subgroup of | C4×D3×SD16, #79649, SD16×D12, … |
| maximal quotient of | C2×#4758, C2×#4761, C2×#4760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1317);