Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C24×#1020 (order 1536) · Inn(G) = C22 · Out(G) = C24×S4 |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C8 | C22 |
| commutator | C2 | C22×C3×C8 |
| Frattini, cosocle | C2×C4 | C23×C3 |
| socle, prime radical | C22×C3 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C8 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,5,6,7,8,9,10,11);(12,13,14,15)(16,19,18,17);(12,16,14,18)(13,17,15,19)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 878);