Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #930 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ #471 |
104 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 39 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)(4,6)(5,10)(7,13)(8,14)(9,16)(11,19)(12,15)(17,18);(4,5,7,9,8,11,15,17)(6,10,13,16,14,19,12,18);(2,3)(5,9)(6,12)(7,15)(11,17)(13,14)(16,18)Polycyclic presentation:
2^6,3|b^2=e,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^2,d^c=df36 conjugacy classes — view the full character table.
| maximal subgroup of | #79892, #79893, #79900, … |
| maximal normal subgroup of | #79892, #79893, #79900, … |
| maximal quotient of | #69891, #69892, #69893, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 700);