Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#3109 (order 768) · Inn(G) = #832 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #360 |
58 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)(4,5)(6,13)(7,16)(8,10)(9,17)(11,15)(12,18)(14,19);(2,3)(5,9,10,18)(6,12)(7,15)(11,13,16,19)(14,17)Polycyclic presentation:
2^6,3|a^2=c,d^2=e,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=g^2,c^b=cf,d^b=def,e^b=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #69246, #69247, D3×#360, … |
| maximal normal subgroup of | #69246, #69247, D3×#360, … |
| maximal quotient of | #67726, #67762, #67764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 125);