Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #213 |
| automorphism | Aut(G) = D3×#7630939 (order 3072) · Inn(G) = #67 · Out(G) = C22×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #427 |
114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(4,5,6,7)(9,14)(10,17)(11,19)(13,20)(15,16)(21,22);(8,10)(9,13)(11,18)(12,17)(14,22)(15,20)(16,21)(19,23);(2,3)(8,9,11,15,12,16,19,14)(10,13,18,20,17,21,23,22)Polycyclic presentation:
2^6,3|a^2=e,c^2=d,e^2=f,c^a=ce,g^a=g^2,c^b=cf,e^c=ef48 conjugacy classes — view the full character table.
| maximal subgroup of | #68699, #68710, #68738, … |
| maximal normal subgroup of | #68699, #68710, #68738, … |
| maximal quotient of | #69468, C4×#808, C4×#807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 557);