Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#942 |
| Fitting | C22×C3×Q8 ⋊ C2 |
| 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)(6,9,10,19)(7,12,14,21)(8,15,16,18)(11,17,13,20);(1,2)(6,8,10,16)(7,11,14,13)(9,18,19,15)(12,20,21,17);(4,5)(6,7)(8,11)(9,17)(10,14)(12,18)(13,16)(15,21)(19,20);(1,2)42 conjugacy classes — view the full character table.
| maximal subgroup of | #69693, #69694, #70110, … |
| maximal normal subgroup of | #69693, #69694, #70110, … |
| maximal quotient of | C2×#4964, C2×#4966, C2×#4967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1367);