Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #937 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#137 ⋊ C2 |
| Sylow | C3 ⋊ #455 |
152 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 152 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)(4,6,8,14)(5,9,10,18)(7,17,15,13)(11,19,16,12);(4,5)(6,12)(7,16)(8,10)(9,13)(11,15)(14,19)(17,18);(2,3)(7,15)(9,18)(11,16)(13,17)Polycyclic presentation:
2^6,3|c^2=f,e^2=f,c^a=cd,e^a=ef,g^a=g^2,c^b=ce,d^b=df,e^b=ef,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#455, #80292, #80298, … |
| maximal normal subgroup of | D3×#455, #80292, #80298, … |
| maximal quotient of | #70298, #70305, #70306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 761);