Finite Group DatabasebetaFinite Group Databasebeta| order | 1000 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C52 · Schur multiplier (H2) = C2×C52 |
| direct product | D4×H(3,𝔽5) |
| unambiguous semidirect product | C4⋊(C2×H(3,𝔽5)) |
| unambiguous semidirect product | C22⋊(C2×H(3,𝔽5)) |
| unambiguous semidirect product | (C52×D4)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2×C5 | C22×C52 |
| LCS | G ▷ C2×C5 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 100 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4)(5,6,8,12,19)(7,10,14,21,26)(9,15,24,29,25)(11,17,23,20,27)(13,22,28,18,16);(2,4)(6,9,16,17,10)(8,13,14,15,23)(12,20,24,21,22)(19,26,27,28,29)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1000, 81);