Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1710 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,20)(16,22,24,29)(17,25,26,21)(19,27,28,23)(30,31)(32,38)(33,35)(34,42)(36,40)(37,41)(39,44)(43,45);(14,16,17,23,18,24,26,27)(15,19,21,22,20,28,25,29)(31,35)(36,43)(37,42)(38,44);(15,20)(19,28)(21,25)(22,29)(30,32,34,40,33,39,41,45)(31,36,37,38,35,43,42,44)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 442);