Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×C4×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C22 |
| commutator | C2 | C23×C3×C4 |
| Frattini, cosocle | C22 | C24×C3 |
| socle | C23×C3 | C23 |
| prime radical | C22×C3×D4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
426 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 13 points:
(1,2)(11,12,13);(3,4,5,6);(7,8,9,10);(8,10)120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1404);