Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#2229 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C13 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ C4×C8×C13 ◁ G | |
124 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 124 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) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,19)(16,20,21,18)(22,23,25,28)(24,32,33,50)(26,38,36,51)(27,41,37,31)(29,45,43,34)(30,47,44,40)(35,52,42,48)(39,49,53,46);(14,16,17,21)(15,18,19,20)(23,28)(29,43)(30,44)(31,41)(32,50)(38,51)(46,49)(48,52);(22,24,25,33)(23,29,28,43)(26,39,36,53)(27,42,37,35)(30,48,44,52)(31,49,41,46)(32,51,50,38)(34,47,45,40)377 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 961);