Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 294912 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C25×C4 |
| abelian | C22×C3×C42 |
| multiplicative | (ℤ/520)ˣ, (ℤ/560)ˣ, (ℤ/624)ˣ, (ℤ/720)ˣ, (ℤ/780)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22 | C24×C3 |
| socle, prime radical | C24×C3 | C22 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2,3,4)(13,14,15);(5,6,7,8);(9,10);(11,12)192 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1400);