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×#918 |
| Fitting | C2×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#137 |
258 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(1,2)(3,4,5)(6,8,10,14)(7,11,13,20)(9,18,19,16)(12,21,17,15);(1,2)(6,7,10,13)(8,15,14,21)(9,17,19,12)(11,18,20,16);(4,5)(8,14)(11,20)(15,21)(16,18);(1,2)36 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#918, #79719, #79730, … |
| maximal normal subgroup of | C4×#918, #79719, #79730, … |
| maximal quotient of | C2×#4817, C2×#4789, C2×#4791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1323);