Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C29 |
| direct product | C13×#3584 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C25×C13 |
| socle | C2×C13 | C23×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C13×#41 ◁ G | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,25)(17,22)(18,23)(19,32)(20,24)(21,27)(26,39)(28,35)(29,37)(30,38)(31,42)(33,36)(34,41)(40,45)(43,44);(14,16)(15,21)(17,26)(18,27)(19,33)(20,28)(22,34)(23,35)(24,36)(25,32)(29,40)(30,41)(31,43)(37,44)(38,45)(39,42);(14,17)(15,22)(16,26)(18,30)(19,31)(20,29)(21,34)(23,38)(24,37)(25,39)(27,41)(28,40)(32,42)(33,43)(35,45)(36,44);(17,29)(22,37)(26,40)(30,31)(34,44)(38,42)(39,45)(41,43);(14,18,20,19)(15,23,24,32)(16,27,28,33)(17,30,29,31)(21,35,36,25)(22,38,37,42)(26,41,40,43)(34,45,44,39)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2316);