Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = (C2×D3)⋊C4 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #324 |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,6,8,7,9,10,11)(12,14)(13,16,17,18)(15,19);(2,3)(12,13,15,17)(14,18,19,16)Polycyclic presentation:
2^6,3|a^2=f,b^2=c,c^2=e,b^a=bd,c^a=cf,g^a=g^2,d^b=df42 conjugacy classes — view the full character table.
| maximal subgroup of | #68100, #68102, #68105, … |
| maximal normal subgroup of | #68100, #68102, #68105, … |
| maximal quotient of | #67446, #67447, #67452, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 28);