Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) = C2×#72216213 (order 12288) · Inn(G) = C24 · Out(G) = C2×#4625⋊C2 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#543 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C24×C3 |
| prime radical | C24×C3 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ G | |
242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18,19);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11,12,16)(10,13,14,15);(2,6)(5,7)66 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1432);