Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×B(4,𝔽2) (order 768) · Inn(G) = #937 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×#123 ⋊ C2 |
| Sylow | C3 ⋊ #458 |
160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 160 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)(4,7,8,14)(5,10,11,18)(6,12,13,19)(9,15,16,17);(4,6)(5,9)(7,12)(8,13)(10,17)(11,16)(14,19)(15,18);(2,3)(4,5)(6,9)(7,10)(8,11)(12,17)(13,16)(14,18)(15,19)Polycyclic presentation:
2^6,3|c^2=f,c^a=cd,e^a=ef,g^a=g^2,c^b=ce,d^b=df30 conjugacy classes — view the full character table.
| maximal subgroup of | #69138, #69139, #69140, … |
| maximal normal subgroup of | #69138, #69139, #69140, … |
| maximal quotient of | #69536, #69538, #69539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 612);