Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#3261 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C23×C13 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
322 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 168 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) = 832 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(22,23,27,31)(24,28,33,42)(25,35,37,45)(26,38,40,30)(29,39,44,36)(32,46,47,53)(34,50,49,43)(41,51,52,48);(14,15)(16,19)(17,18)(20,21)(22,24,27,33)(23,28,31,42)(25,32,37,47)(26,34,40,49)(29,41,44,52)(30,43,38,50)(35,46,45,53)(36,48,39,51);(15,18)(19,21)(22,25,27,37)(23,29,31,44)(24,32,33,47)(26,39,40,36)(28,41,42,52)(30,45,38,35)(34,51,49,48)(43,53,50,46);(14,16,17,20)(15,19,18,21)(24,33)(28,42)(32,47)(34,49)(41,52)(43,50)(46,53)(48,51)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1993);