Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C27×D3 (order 768) · Inn(G) = C22×D3 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #67⋊C8 |
| Fitting | C3×C22⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ C8×D4 |
134 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 61 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)(5,7)(8,9,10,12,11,13,14,15);(4,5)(6,7);(2,3)(5,7)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,b^a=be,g^a=g^2,c^b=ce60 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67⋊C8, #78596, #78601, … |
| maximal normal subgroup of | C2×#67⋊C8, #78596, #78601, … |
| maximal quotient of | #68148, #68149, #68150, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 284);