Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C29 |
| direct product | C13×#3586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C25×C13 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C13×#41 ◁ G | |
1412 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)(16,24)(17,21)(18,30)(19,22)(20,26)(23,32)(25,38)(27,37)(28,35)(29,42)(31,34)(33,40)(36,41)(39,45)(43,44);(14,16,19,27)(15,20,22,34)(17,25,28,39)(18,31,32,26)(21,33,35,44)(23,37,30,24)(29,43,41,40)(36,45,42,38);(14,17)(15,21)(16,25)(18,29)(19,28)(20,33)(22,35)(23,36)(24,38)(26,40)(27,39)(30,42)(31,43)(32,41)(34,44)(37,45);(17,28)(21,35)(25,39)(29,41)(33,44)(36,42)(38,45)(40,43);(15,22)(20,34)(21,35)(23,30)(24,37)(33,44)(36,42)(38,45)533 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2318);