Finite Group DatabasebetaFinite Group Databasebeta| order | 1392 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C29 · Schur multiplier (H2) = C1 |
| direct product | C3×SD16×C29 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C29 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C3×C29 |
| prime radical | C3×D4×C29 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C29 ◁ C3×C4×C29 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 696 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3)(4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40);(5,7)(6,10)(9,11)609 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1392, 87);