Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #937 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×AGL(1,ℤ/8) ⋊ C2 |
| Sylow | C3 ⋊ #453 |
168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 168 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) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2,3)(5,8)(6,11)(9,10);(4,5)(6,8)(7,9)(10,11);(2,3)(6,11)(8,10)Polycyclic presentation:
2^6,3|e^2=f,c^a=cd,e^a=ef,g^a=g^2,c^b=ce,d^b=df,e^b=ef,e^c=ef27 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#453, #80278, #80284, … |
| maximal normal subgroup of | D3×#453, #80278, #80284, … |
| maximal quotient of | #70297, #70305, #70306, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 757);