Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1718 |
| 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 | |
164 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(14,15,18,21)(16,22,23,29)(17,25,26,20)(19,27,28,24)(30,31)(32,38)(33,35)(34,42)(36,40)(37,41)(39,44)(43,45);(14,16,17,24,18,23,26,27)(15,19,20,22,21,28,25,29)(31,35)(36,43)(37,42)(38,44);(16,23)(19,28)(22,29)(24,27)(30,32,34,40,33,39,41,45)(31,36,37,38,35,43,42,44)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 450);