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) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #430 |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,15,9,16,8,17)(5,11,12,14,10,18,13,19);(5,10)(11,18)(12,13)(14,19);(2,3)(4,5,8,13,9,10,7,12)(6,14,15,18,16,19,17,11)Polycyclic presentation:
2^6,3|a^2=e,c^2=d,d^2=f,e^2=f,b^a=bf,c^a=ce,g^a=g^2,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #68706, #68709, #68722, … |
| maximal normal subgroup of | #68706, #68709, #68722, … |
| maximal quotient of | #69873, #69875, #69983, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 683);