Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C22×D3×AGL(1,ℤ/8) (order 768) · Inn(G) = D12 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C4×C8 ⋊ C2 |
| Sylow | C3 ⋊ #443 |
106 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)(4,7,8,17,10,16,9,6)(5,12,13,19,14,11,18,15);(4,6,9,16,10,17,8,7)(5,11,13,15,14,12,18,19);(2,3)(4,5)(6,15)(7,12)(8,13)(9,18)(10,14)(11,16)(17,19)Polycyclic presentation:
2^6,3|b^2=e,c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #67441, #67443, #67568, … |
| maximal normal subgroup of | #67441, #67443, #67568, … |
| maximal quotient of | C8×D24, #68052, C8×#204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 259);