Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) order 6144 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C2×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ #418 |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(4,5,6,7)(8,9,10,12,11,13,14,15);(9,13)(12,15);(2,3)(9,12)(10,14)(13,15)Polycyclic presentation:
2^6,3|c^2=de,d^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #68713, #68726, #68760, … |
| maximal normal subgroup of | #68713, #68726, #68760, … |
| maximal quotient of | #69812, #69823, #69828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 690);