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) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #356 |
60 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)(4,6,7,13,9,15,5,10)(8,14,11,12,17,18,16,19);(2,3)(4,5,9,7)(6,12,15,19)(8,16,17,11)(10,18,13,14)Polycyclic presentation:
2^6,3|a^2=f,b^2=c,c^2=f,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 | #69161, #69162, #69164, … |
| maximal normal subgroup of | #69161, #69162, #69164, … |
| maximal quotient of | #67453, #67455, #67457, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 37);