Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#889 |
| Sylow | C3 ⋊ C2×#125 |
226 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 226 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) = 21 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5)(6,8,9,12)(7,10,11,13)(15,19)(18,21);(8,12)(10,13)(14,16,17,20)(15,18,19,21);(4,5)(6,7,9,11)(8,10,12,13)(14,15)(16,18)(17,19)(20,21);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68128, C4×#889, #73631, … |
| maximal normal subgroup of | #68128, C4×#889, #73631, … |
| maximal quotient of | C2×#4650, C2×#4652, C2×#4653, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1057);