Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2123 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C2×#99 |
| commutator | C23 | C22×C4×C13 |
| Frattini, cosocle | C2×D4 | C23×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 416 |
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)(16,24)(17,21)(18,31,32,22)(19,23)(20,26,34,41)(25,38)(27,39)(28,42)(29,44,45,36)(30,37)(33,40,35,43);(14,16,19,27)(15,20,23,34)(17,28,30,25)(18,26,32,41)(21,35,37,33)(22,24,31,39)(29,43,45,40)(36,42,44,38);(14,17,19,30)(15,21,23,37)(16,25,27,28)(18,29,32,45)(20,33,34,35)(22,36,31,44)(24,38,39,42)(26,40,41,43)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 855);