Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×D4 (order 384) · Inn(G) = C2×#67 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ #444 |
104 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 47 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)(4,7,8,11,10,18,9,17)(5,6,12,14,13,15,16,19);(4,6)(5,11)(7,16)(8,14)(9,19)(10,15)(12,18)(13,17);(2,3)(4,5)(6,7)(8,12)(9,16)(10,13)(11,14)(15,18)(17,19)Polycyclic presentation:
2^6,3|c^2=d,d^2=f,e^2=f,b^a=be,e^a=ef,g^a=g^2,c^b=cf,e^b=ef42 conjugacy classes — view the full character table.
| maximal subgroup of | #79889, #79890, D3×#444, … |
| maximal normal subgroup of | #79889, #79890, D3×#444, … |
| maximal quotient of | #69888, #69890, #69899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 704);