Finite Group DatabasebetaFinite Group Databasebeta| order | 736 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C23 · Schur multiplier (H2) = C25 |
| direct product | C23×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C23 | C24 |
| commutator, Frattini, cosocle | C2 | C24×C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C23 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 184 |
A minimal faithful permutation representation — 4 generators on 31 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)(26,30)(27,28)(29,31);(25,28)(29,30);(24,25)(26,29)(27,28)(30,31);(24,26)(25,29)(27,31)(28,30)391 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(736, 192);