Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 147456 |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22 |
| commutator, Frattini, cosocle | C2 | C25×C3 |
| socle | C23×C3 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 15 points:
(1,2)(5,6,7);(3,4);(9,12)(13,15);(8,9)(10,13)(11,12)(14,15);(1,2)(3,4)(8,10,11,14)(9,13,12,15)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1533);