Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#951 |
| Sylow | C3 ⋊ C2×#137 |
258 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 111 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:
(3,4,5)(6,8)(7,11)(9,18)(10,15)(12,21)(13,20)(14,17)(16,19);(1,2)(8,15)(11,20)(14,21)(16,18);(4,5)(6,7,10,13)(8,14,15,21)(9,17,19,12)(11,18,20,16);(1,2)42 conjugacy classes — view the full character table.
| maximal subgroup of | #70004, #70007, #70008, … |
| maximal normal subgroup of | #70004, #70007, #70008, … |
| maximal quotient of | C2×#4995, C2×#5004, C2×#4996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1382);