Finite Group DatabasebetaFinite Group Databasebeta| order | 736 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C23 · Schur multiplier (H2) = C2 |
| direct product | C23×#126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C23 | C23 |
| commutator, Frattini, cosocle | C22 | C23×C23 |
| prime radical | C23×C23 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C23 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 368 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23)(25,29)(26,33)(31,38)(32,36)(34,37)(35,39);(24,25,30,36)(26,31,37,38)(27,29,28,32)(33,39,34,35);(24,26,27,34)(25,31,29,35)(28,33,30,37)(32,39,36,38)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(736, 161);