Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) = C2×#72224760 (order 3072) · Inn(G) = C42 · Out(G) = C2×GL(2,ℤ/4) |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#338 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C42 |
| commutator | C4 | C3×C42 |
| Frattini, cosocle | C22×C4 | C22×C3 |
| socle | C2×C3 | #95 |
| prime radical | C23×C3 | Q8 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,4,8,6,9,5,14)(3,10,12,7,13,15,11,16)(17,18,19);(1,3,5,12,6,13,4,11)(2,7,8,10,9,16,14,15)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 144);