Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#2020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ G | |
342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,17)(18,19,24,29)(20,30,34,41)(21,35,39,44)(22,28,23,27)(25,33,42,32)(26,38,45,37)(31,46,48,40)(36,43,47,49);(15,17)(18,20)(19,25)(21,36)(22,33)(23,32)(24,34)(26,43)(27,41)(28,30)(29,42)(31,38)(35,46)(37,48)(39,47)(40,44)(45,49);(18,21,24,39)(19,26,29,45)(20,31,34,48)(22,38,23,37)(25,40,42,46)(27,44,28,35)(30,43,41,49)(32,36,33,47)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 752);