Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#2988 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C24 |
| commutator | C2 | C23×C8×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C22×C13 | C23×C4 |
| prime radical | C2×C13×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
432 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(22,23,24,26,25,27,28,29);(14,15)(16,19)(17,18)(20,21);(15,18)(19,21);(14,16,17,20)(15,19,18,21)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1720);