Finite Group DatabasebetaFinite Group Databasebeta| order | 1750 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | Aut(G) = C2×C3×#92539 (order 3000) · Inn(G) = C52 · Out(G) = C2×C3×AGL(1,𝔽5) |
| homology | Abelianization (H1) = C2×C52×C7 · Schur multiplier (H2) = C1 |
| direct product | C2×C7×5−1+2 |
| unambiguous semidirect product | (C2×C25)⋊(C5×C7) |
| unambiguous semidirect product | (C2×C7×C25)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5×C7 | C52 |
| commutator, Frattini, cosocle | C5 | C2×C52×C7 |
| prime radical | C2×C52×C7 | C5 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C7 ◁ G | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 350 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,2)(3,4,5,6,7,8,9)(10,11,13,17,24,12,15,19,26,31,16,22,28,25,32,23,21,18,27,33,30,14,20,29,34);(11,14,21,22,15)(13,18,19,20,28)(17,25,29,26,27)(24,31,32,33,34)406 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1750, 22);