Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#885 |
| Fitting | C2×C3×C22⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#124 |
330 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 119 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)(7,9)(10,12)(11,15)(13,17)(14,16);(6,7,8,9)(11,14)(12,17);(4,5)(7,9)(10,11)(12,16)(13,14)(15,17);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68104, C4×#885, #73452, … |
| maximal normal subgroup of | #68104, C4×#885, #73452, … |
| maximal quotient of | C2×#4650, C2×#4669, C2×#4670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1050);