Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#2955 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C2×D4 |
| commutator | C4 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C2×C13 | C2×C4×D4 |
| prime radical | C13×#575 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ G | |
456 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)(14,15,19,24,20,23,18,32)(16,25,27,21,28,36,33,35)(17,22,30,38,31,37,29,42)(26,39,40,34,41,45,43,44);(14,16)(15,21)(17,26)(18,33)(19,27)(20,28)(22,34)(23,35)(24,36)(25,32)(29,43)(30,40)(31,41)(37,44)(38,45)(39,42);(14,17,19,30,20,31,18,29)(15,22,24,38,23,37,32,42)(16,26,27,40,28,41,33,43)(21,34,36,45,35,44,25,39);(15,23)(21,35)(22,37)(24,32)(25,36)(34,44)(38,42)(39,45)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1687);