Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C3 ⋊ #419 |
150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,11,13,20)(9,15,16,23)(10,17,12,19)(14,22,21,18);(8,10,13,12)(9,14,16,21)(11,19,20,17)(15,18,23,22);(2,3)(4,5,6,7)(8,9)(10,14)(11,18)(12,21)(13,16)(15,17)(19,23)(20,22)Polycyclic presentation:
2^6,3|a^2=d,b^2=f,c^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 | #69673, #69693, #70106, … |
| maximal normal subgroup of | #69673, #69693, #70106, … |
| maximal quotient of | #71553, #71558, #71621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 784);