Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×#72209832 (order 12288) · Inn(G) = C22 · Out(G) = C22×#87588⋊C2 |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C22×C3×M16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22 |
| commutator | C2 | C23×C3×C4 |
| Frattini, cosocle | C4 | C24×C3 |
| socle | C23×C3 | C2×C4 |
| prime radical | C24×C3 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(13,14,15);(3,4);(5,6,7,8,9,10,11,12);(6,10)(8,12)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1455);