Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = C2×D3×D4 · Out(G) = C23 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D3×#137 |
| Fitting | C3×#137 ⋊ C2 |
| Sylow | C3 ⋊ C2×#137 |
258 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 4 generators on 19 points:
(2,7)(6,15)(8,12)(9,16)(17,18,19);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9);(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15);(18,19)33 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#455, #80293, D3×#456, … |
| maximal normal subgroup of | D3×#455, #80293, D3×#456, … |
| maximal quotient of | #78867, #78870, #78878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1335);