Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#61451 (order 1536) · Inn(G) = #832 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #361 |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(5,9)(6,14)(7,17)(10,11)(12,19)(13,16)(15,18);(2,3)(4,5,6,10,8,13,14,12)(7,16,15,11,18,9,17,19)Polycyclic presentation:
2^6,3|a^2=c,c^2=f,d^2=e,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,c^b=cf,d^b=def,e^b=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#361, #69256, #69257, … |
| maximal normal subgroup of | D3×#361, #69256, #69257, … |
| maximal quotient of | #67723, #67762, #67775, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 122);