Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #882 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#104 ⋊ C2 |
| Sylow | C3 ⋊ #456 |
142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,8,6,10,14,9,15)(5,11,12,16,13,19,17,18);(4,6,10,15)(5,11,13,19)(7,8,14,9)(12,16,17,18);(2,3)(4,5)(6,11)(7,16)(8,17)(9,12)(10,13)(14,18)(15,19)Polycyclic presentation:
2^6,3|b^2=f,c^2=d,d^2=f,e^2=f,c^a=cd,d^a=df,g^a=g^2,c^b=ce,e^b=ef,e^c=ef30 conjugacy classes — view the full character table.
| maximal subgroup of | #78986, C2×#4826, #78995, … |
| maximal normal subgroup of | #78986, C2×#4826, #78995, … |
| maximal quotient of | #68184, #68185, #68192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 385);