Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = C24×D3×D4 (order 768) · Inn(G) = #67 · Out(G) = C25 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D4⋊(C3⋊C8) |
| Sylow | C3 ⋊ D4⋊C8 |
82 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(13,16)(14,19)(17,18);(2,3)(4,5,6,8,7,9,10,11)(12,13)(14,16)(15,18)(17,19)Polycyclic presentation:
2^6,3|a^2=c,c^2=e,d^2=f,b^a=bd,d^a=df,g^a=g^2,d^b=df48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4⋊C8, #68165, #68166, … |
| maximal normal subgroup of | D3×D4⋊C8, #68165, #68166, … |
| maximal quotient of | #67619, #67638, #67639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 93);