Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = D3×#394102 (order 3072) · Inn(G) = D12 · Out(G) = C2×#392 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #341 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(4,5,6,7)(8,10,11,18,13,20,12,19)(9,14,21,23,16,22,15,17);(2,3)(8,9,11,15,13,16,12,21)(10,17,18,22,20,23,19,14)Polycyclic presentation:
2^6,3|a^2=c,b^2=de,c^2=f,d^2=f,b^a=bd,d^a=df,g^a=g^260 conjugacy classes — view the full character table.
| maximal subgroup of | #67527, #67529, #67775, … |
| maximal normal subgroup of | #67527, #67529, #67775, … |
| maximal quotient of | #67654, #67656, #67679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 111);