Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of #5386 |
| automorphism | Aut(G) = C24×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,4)(5,13,18,6,8,20,10,11,16,7,17,19)(9,14,15,12);(1,2)(3,4)(5,6,10,7)(8,12,17,14)(9,11,15,13)(16,20,18,19)Polycyclic presentation:
2^3,3,2^3|a^2=g,b^2=g,e^2=g,f^2=g,b^a=bc,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg23 conjugacy classes — view the full character table.
| maximal subgroup of | #85335, #85338, #85343, … |
| maximal normal subgroup of | #85335, #85338, #85343, … |
| maximal quotient of | #73059, #73061, #73089, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 984);