Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #884 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#100 ⋊ C2 |
| Sylow | C3 ⋊ #471 |
104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 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)(5,10)(6,12)(7,14)(9,15)(11,17)(16,19);(4,6,7,13,8,12,14,18)(5,9,10,15,11,16,17,19);(2,3)(4,5)(6,9)(7,10)(8,11)(12,16)(13,15)(14,17)(18,19)Polycyclic presentation:
2^6,3|b^2=d,d^2=f,c^a=cde,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,d^c=df27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4748, #78703, #78719, … |
| maximal normal subgroup of | C2×#4748, #78703, #78719, … |
| maximal quotient of | #68394, #68397, #68398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 307);