Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×#2196 (order 1536) · Inn(G) = #882 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#101 ⋊ C2 |
| Sylow | C3 ⋊ #456 |
142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,8,17,9,18,19,10)(5,11,12,16,13,6,14,15);(4,6,9,11)(5,10,13,17)(7,12,18,14)(8,16,19,15);(2,3)(4,5)(6,10)(7,16)(8,14)(9,13)(11,17)(12,19)(15,18)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,d^2=f,c^a=cd,d^a=df,e^a=ef,g^a=g^2,c^b=ce,d^b=df,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | #69161, #69162, #69167, … |
| maximal normal subgroup of | #69161, #69162, #69167, … |
| maximal quotient of | #68132, #68133, #68134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 311);