Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×D42 (order 768) · Inn(G) = #884 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#101 ⋊ C2 |
| Sylow | C3 ⋊ #471 |
104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 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)(4,7,9,16)(5,10,12,13)(6,11,14,19)(8,17,18,15);(4,6,8,13,9,14,18,10)(5,7,11,15,12,16,19,17);(2,3)(4,5)(6,7)(8,11)(9,12)(10,17)(13,15)(14,16)(18,19)Polycyclic presentation:
2^6,3|b^2=d,c^2=f,d^2=f,c^a=cde,e^a=ef,g^a=g^2,c^b=ce,e^b=ef,d^c=df27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4754, #78712, #78719, … |
| maximal normal subgroup of | C2×#4754, #78712, #78719, … |
| maximal quotient of | #68440, #68442, #68446, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 313);