Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C3×#379 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C23 |
| commutator | C22 | C22×C3×C4 |
| Frattini, cosocle | C23 | C23×C3 |
| prime radical | C25×C3 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15);(6,8);(10,12)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 812);