Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #895 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#108 ⋊ C2 |
| Sylow | C3 ⋊ #471 |
104 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,8,17,9,18,19,10)(5,11,12,16,14,6,13,15);(4,6,8,15,9,11,19,16)(5,10,13,18,14,17,12,7);(2,3)(4,5)(6,10)(7,16)(8,13)(9,14)(11,17)(12,19)(15,18)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=cef,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4898, #79256, #79261, … |
| maximal normal subgroup of | C2×#4898, #79256, #79261, … |
| maximal quotient of | #68536, #68537, #68544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 457);