Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C2×D3×B(4,𝔽2) (order 768) · Inn(G) = D3×C22⋊C4 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C3×#99 ⋊ C2 |
| Sylow | C3 ⋊ #410 |
158 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 158 conjugacy classes of subgroups — too many to draw. Only its 51 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)(5,10)(6,12,13,14)(8,15)(9,16,17,18)(11,19);(4,6)(5,9)(7,14)(8,13)(10,18)(11,17)(12,15)(16,19);(2,3)(4,5,8,11)(6,9,13,17)(7,10,15,19)(12,16,14,18)Polycyclic presentation:
2^6,3|a^2=f,c^2=d,g^a=g^2,c^b=ce,d^b=df,e^c=ef33 conjugacy classes — view the full character table.
| maximal subgroup of | #69136, #69137, #69144, … |
| maximal normal subgroup of | #69136, #69137, #69144, … |
| maximal quotient of | #68100, #68101, #68102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 299);