Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#6628842 (order 3072) · Inn(G) = D3×D4 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #441 |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(8,10,12,16)(9,13,15,22)(11,20,21,18)(14,23,19,17);(8,9,12,15)(10,17,16,23)(11,19,21,14)(13,20,22,18);(2,3)(4,5,6,7)(10,16)(13,22)(17,23)(18,20)Polycyclic presentation:
2^6,3|a^2=d,b^2=f,c^2=f,e^2=f,c^a=cf,g^a=g^2,c^b=ce,e^b=ef,e^c=ef36 conjugacy classes — view the full character table.
| maximal subgroup of | #69264, #71521, #79708, … |
| maximal normal subgroup of | #69264, #71521, #79708, … |
| maximal quotient of | #70220, #70223, #70238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 743);