Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = D3×#63011 (order 1536) · Inn(G) = C2×#67 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ #421 |
154 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,6,7,15,9,14,8,17)(5,10,11,19,12,18,16,13);(6,14)(10,18)(13,19)(15,17);(2,3)(4,5)(6,13)(7,11)(8,16)(9,12)(10,15)(14,19)(17,18)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,e^2=f,c^a=ce,e^a=ef,g^a=g^2,c^b=cf,e^c=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #68705, #68708, #68725, … |
| maximal normal subgroup of | #68705, #68708, #68725, … |
| maximal quotient of | #69798, #69799, #69801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 698);