Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = D3×D4 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | D3×#103 |
| Fitting | C3×#103 ⋊ C2 |
| Sylow | C3 ⋊ C2×#103 |
162 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 15 points:
(5,6,8,10)(7,12,11,9)(13,14,15);(1,2,3,4)(6,9)(7,11)(10,12);(14,15)42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×#103, #78790, #78805, … |
| maximal normal subgroup of | C2×D3×#103, #78790, #78805, … |
| maximal quotient of | D3×Q8⋊C8, #68232, #68236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 360);