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 ⋊ #352 |
68 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)(7,15,16,13)(8,14)(10,18,19,12)(11,17);(2,3)(4,5,8,11)(6,12,14,18)(7,10,16,19)(9,15,17,13)Polycyclic presentation:
2^6,3|a^2=f,b^2=c,b^a=bd,c^a=ce,g^a=g^2,d^b=de,e^b=ef,d^c=df24 conjugacy classes — view the full character table.
| maximal subgroup of | #69135, #69136, D3×#352, … |
| maximal normal subgroup of | #69135, #69136, D3×#352, … |
| maximal quotient of | #67445, #67447, #67458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 31);