Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×B(4,𝔽2) (order 768) · Inn(G) = #806 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #354 |
70 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)(5,9,10,18)(7,16,17,13)(8,14)(11,19)(12,15);(2,3)(4,5,8,10)(6,12,14,19)(7,15,17,11)(9,16,18,13)Polycyclic presentation:
2^6,3|a^2=f,b^2=c,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=g^2,d^b=def,e^b=ef,d^c=df24 conjugacy classes — view the full character table.
| maximal subgroup of | #69131, #69132, #69138, … |
| maximal normal subgroup of | #69131, #69132, #69138, … |
| maximal quotient of | #67449, #67450, #67452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 32);