Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4451 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C22⋊C8 ⋊ C2 |
| Sylow | C3 ⋊ #327 |
90 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)(4,6)(5,8,9,10)(7,11)(12,13,17,19,15,18,14,16);(2,3)(4,5)(6,10)(7,9)(8,11)(13,16)(14,17)(18,19)Polycyclic presentation:
2^6,3|b^2=cd,c^2=e,b^a=bc,c^a=ce,d^a=df,g^a=g^2,c^b=cf,d^b=df39 conjugacy classes — view the full character table.
| maximal subgroup of | #68106, #68107, #68108, … |
| maximal normal subgroup of | #68106, #68107, #68108, … |
| maximal quotient of | #67448, #67450, #67456, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 29);