Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #939 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| 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)(5,10)(6,14)(7,17)(11,12)(13,18)(15,16);(4,6)(5,9)(7,14)(8,16)(10,13)(11,18)(12,19)(15,17);(2,3)(4,5,8,12)(6,13,16,18)(7,10,17,11)(9,14,19,15)Polycyclic presentation:
2^6,3|a^2=f,d^2=f,e^2=f,b^a=be,c^a=cd,d^a=df,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,d^c=df,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | #80280, #80281, #80288, … |
| maximal normal subgroup of | #80280, #80281, #80288, … |
| maximal quotient of | #70299, #70302, #70304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 759);