Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1704 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
140 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 64 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)(16,22,24,29)(17,26,23,20)(19,27,28,25)(30,31)(32,38)(33,35)(34,42)(36,40)(37,41)(39,45)(43,44);(14,16,23,27,18,24,17,25)(15,19,26,29,21,28,20,22)(31,35)(36,44)(37,42)(38,45);(16,24)(19,28)(22,29)(25,27)(30,32,33,39)(31,36,35,44)(34,43,41,40)(37,45,42,38)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 436);