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×#3407 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C24 |
| commutator | C2 | C23×C8×C13 |
| Frattini, cosocle | C8 | C24×C13 |
| socle | C2×C13 | C23×C8 |
| prime radical | C2×C13×#41 | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,22,18,23,27,35,19,21,28,33,29,34,39,43)(16,20,25,31,26,32,38,42,24,30,36,40,37,41,44,45);(14,16)(15,20)(17,25)(18,26)(19,24)(21,30)(22,31)(23,32)(27,38)(28,36)(29,37)(33,40)(34,41)(35,42)(39,44)(43,45);(16,24)(20,30)(25,36)(26,37)(31,40)(32,41)(38,44)(42,45);(15,21)(20,30)(22,33)(23,34)(31,40)(32,41)(35,43)(42,45)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2139);