Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22 |
| commutator | C2 | C23×C3×C4 |
| Frattini, cosocle | C22 | C24×C3 |
| socle, prime radical | C23×C3 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2)(15,16,17);(3,4,5,6);(7,8,9,10)(11,14,13,12);(7,11,9,13)(8,12,10,14)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1405);