Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#898 |
| Fitting | C2×C3×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ C2×#126 |
246 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 246 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:
(1,2)(3,4,5)(6,8,9,12)(7,11,10,13)(15,19)(18,21);(7,10)(8,12)(14,16,17,20)(15,18,19,21);(4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,18)(17,19)(20,21);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68129, C4×#898, #73629, … |
| maximal normal subgroup of | #68129, C4×#898, #73629, … |
| maximal quotient of | C2×#4651, C2×#4654, C2×#4669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1071);