Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C22×D3×AGL(1,ℤ/8) (order 768) · Inn(G) = D3×D4 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | D3×#135 |
| Fitting | C3×#135 ⋊ C2 |
| Sylow | C3 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D3×D4 |
| commutator, cosocle | C3×C4 | C24 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×#135 | C2 |
| socle | C2×C3 | C22×D4 |
| hypercenter | #135 | D3 |
| hypocenter | C3 | C2×#135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
266 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 4 generators on 19 points:
(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,18,19);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13);(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16);(18,19)42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#359, #69241, D3×#360, … |
| maximal normal subgroup of | D3×#359, #69241, D3×#360, … |
| maximal quotient of | #78818, #78822, #78825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1326);