Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#23900 (order 3072) · Inn(G) = C22×D3 · Out(G) = C2×D42 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#907 |
| Fitting | C2×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#131 |
266 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 151 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)(6,8,9,15,10,14,16,20)(7,12,13,19,11,17,18,21);(1,2)(7,11)(8,14)(13,18)(15,20);(4,5)(6,7)(8,12)(9,13)(10,11)(14,17)(15,19)(16,18)(20,21);(1,2)60 conjugacy classes — view the full character table.
| maximal subgroup of | #68710, #68724, #68739, … |
| maximal normal subgroup of | #68710, #68724, #68739, … |
| maximal quotient of | C2×#4701, #78481, C2×#4708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1303);