Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C25×D3×D4 (order 1536) · Inn(G) = #67 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×#214 |
| Sylow | C3 ⋊ C4×SD16 |
122 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 55 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:
(1,2,3)(9,12)(10,15)(13,14);(2,3)(8,9,11,14)(10,12,15,13);(4,5,6,7)48 conjugacy classes — view the full character table.
| maximal subgroup of | #68149, #68150, #68154, … |
| maximal normal subgroup of | #68149, #68150, #68154, … |
| maximal quotient of | #69460, #69464, C4×#810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 576);