Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#63011 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #421 |
154 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 67 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)(7,15)(8,17)(9,14)(11,19)(12,18)(13,16);(6,14)(10,18)(13,19)(15,17);(2,3)(4,5,7,11,9,12,8,16)(6,13,15,10,14,19,17,18)Polycyclic presentation:
2^6,3|a^2=d,d^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #70005, #70012, #70038, … |
| maximal normal subgroup of | #70005, #70012, #70038, … |
| maximal quotient of | #71574, #71590, #71597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 795);