Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#878 |
| Sylow | C3 ⋊ C2×C22⋊Q8 |
322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(3,4,5)(6,7)(8,9)(10,13,14,12)(11,15,16,17);(1,2)(7,9)(10,12)(11,15)(13,14)(16,17);(4,5)(10,11,14,16)(12,15,13,17);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68955, C4×#878, #73447, … |
| maximal normal subgroup of | #68955, C4×#878, #73447, … |
| maximal quotient of | C2×#4654, C2×#4655, C2×#4647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1040);