Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of D48 |
| automorphism | Aut(G) = C22×D3×#442 (order 1536) · Inn(G) = D24 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #358 |
58 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3)(4,6,7,5)(8,9,14,23,12,16,10,13,11,15,17,22,18,20,19,21);(2,3)(4,5,7,6)(9,13)(10,17)(12,18)(14,19)(15,21)(16,23)(20,22)Polycyclic presentation:
2^6,3|a^2=df,b^2=cde,c^2=ef,e^2=f,b^a=bc,c^a=ce,e^a=ef,g^a=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | #69059, C4×#800, C4×Dic24, … |
| maximal normal subgroup of | #69059, C4×#800, C4×Dic24, … |
| maximal quotient of | #67505, #67528, #67561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 62);