Finite Group DatabasebetaFinite Group Databasebeta| order | 736 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C23 · Schur multiplier (H2) = C1 |
| direct product | C23×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C23 | C22 |
| commutator | C2 | C2×C8×C23 |
| Frattini, cosocle | C8 | C22×C23 |
| socle | C2×C23 | C2×C8 |
| prime radical | C22×C23 | C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C23 ◁ G | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 368 |
A minimal faithful permutation representation — 2 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)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39);(25,33)(27,35)(29,37)(31,39)460 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(736, 59);