Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#7622880 (order 3072) · Inn(G) = #217 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #410 |
158 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)(4,7)(5,10)(6,12)(8,15)(9,16)(11,14)(13,19)(17,18);(4,6,8,13)(5,9,11,17)(7,12,15,19)(10,16,14,18);(2,3)(4,5)(6,9)(7,14,15,10)(8,11)(12,18,19,16)(13,17)Polycyclic presentation:
2^6,3|a^2=d,b^2=f,c^a=ce,e^a=ef,g^a=g^2,d^c=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #67691, #67692, #67696, … |
| maximal normal subgroup of | #67691, #67692, #67696, … |
| maximal quotient of | #71571, #71572, #71592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 799);