Finite Group DatabasebetaFinite Group Databasebeta| order | 1375 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×5−1+2 |
| unambiguous semidirect product | C25⋊(C5×C11) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C11 | C52 |
| commutator, Frattini, cosocle | C5 | C52×C11 |
| prime radical | C52×C11 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C11 ◁ G | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 275 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,19,26,14,17,21,28,33,18,24,30,27,34,25,23,20,29,35,32,16,22,31,36);(13,16,23,24,17)(15,20,21,22,30)(19,27,31,28,29)(26,33,34,35,36)319 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1375, 4);