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) = C22 |
| direct product | Dic3×SD16 |
| Sylow | C3 ⋊ C4×SD16 |
122 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(6,9)(7,12)(10,11)(13,14,15);(5,6,8,11)(7,9,12,10);(1,2,3,4)(14,15)42 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×Dic6, #79632, #79633, … |
| maximal normal subgroup of | SD16×Dic6, #79632, #79633, … |
| maximal quotient of | Dic3×#102, #70156, #70170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 720);