Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×#11713200 (order 3072) · Inn(G) = C22 · Out(G) = C22×D4×S4 |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#131 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22 |
| commutator | C2 | C23×C3×C4 |
| Frattini, cosocle | C4 | C24×C3 |
| socle | C22×C3 | C22×C4 |
| prime radical | C2×C3×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2)(3,4,5);(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21);(8,14)(11,17)(15,20)(18,21);(1,2)(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1456);