Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = C23×D3×S4 (order 1152) · Inn(G) = C22×D3 · Out(G) = C23×D3 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | D3×#131 |
| Fitting | C3×#131 ⋊ C2 |
| Sylow | C3 ⋊ C2×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D3 |
| commutator, socle | C2×C3 | C23×C4 |
| Frattini | C4 | C23×D3 |
| Fitting | C3×#131 | C2 |
| prime radical | D3×#41 | C2 |
| cosocle | C3×C4 | C24 |
| hypercenter | #131 | D3 |
| hypocenter | C3 | C2×#131 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ #131 | |
266 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 149 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:
(3,9)(6,12)(10,15)(13,16)(17,18,19);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16);(18,19)60 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#350, #69125, D3×#443, … |
| maximal normal subgroup of | D3×#350, #69125, D3×#443, … |
| maximal quotient of | #78542, #78544, D3×#405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1308);