Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#511819 (order 3072) · Inn(G) = C2×D12 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #430 |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,8,16,9,14,7,15)(5,10,11,13,12,18,17,19);(6,14)(10,18)(13,19)(15,16);(2,3)(4,5,7,11,9,12,8,17)(6,13,15,18,14,19,16,10)Polycyclic presentation:
2^6,3|a^2=d,c^2=e,d^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf42 conjugacy classes — view the full character table.
| maximal subgroup of | #68701, #68715, #68718, … |
| maximal normal subgroup of | #68701, #68715, #68718, … |
| maximal quotient of | #69840, #69841, #69842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 684);