Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×D4×AGL(1,ℤ/8) (order 1536) · Inn(G) = #938 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×Q16 ⋊ C2 |
| Sylow | C3 ⋊ #473 |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(4,7,10,23)(5,12,14,29)(6,15,18,32)(8,22,9,21)(11,25,27,34)(13,30,24,19)(16,20,17,31)(26,35,33,28);(4,6,9,17,10,18,8,16)(5,11,13,26,14,27,24,33)(7,20,22,32,23,31,21,15)(12,28,19,34,29,35,30,25);(2,3)(4,5)(6,11)(7,19)(8,24)(9,13)(10,14)(12,21)(15,28)(16,33)(17,26)(18,27)(20,34)(22,29)(23,30)(25,31)(32,35)Polycyclic presentation:
2^6,3|b^2=e,c^2=f,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,d^c=df,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69270, #69271, #71490, … |
| maximal normal subgroup of | #69270, #69271, #71490, … |
| maximal quotient of | #70263, #70264, #70265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 751);