Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #4466 |
| automorphism | Aut(G) = D3×#511819 (order 3072) · Inn(G) = C4×Dic3 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ H(3,ℤ/4) |
94 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)(4,6,7,15,9,14,8,16)(5,10,11,13,12,18,17,19);(2,3)(4,5)(6,13,14,19)(7,17)(8,11)(9,12)(10,16,18,15)Polycyclic presentation:
2^6,3|a^2=c,b^2=d,d^2=f,e^2=f,b^a=be,d^a=df,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #67534, #67536, #67778, … |
| maximal normal subgroup of | #67534, #67536, #67778, … |
| maximal quotient of | #67660, #67666, #67676, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 116);