Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#1858 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #120 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #385 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
252 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 108 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)(14,15,19,23,20,24,18,31)(16,25,27,21,28,36,32,35)(17,22,29,38,30,39,33,37)(26,41,44,45,43,40,42,34);(14,16,19,27,20,28,18,32)(15,21,23,36,24,35,31,25)(17,26,29,44,30,43,33,42)(22,34,38,41,39,45,37,40);(14,17)(15,22)(16,26)(18,33)(19,29)(20,30)(21,34)(23,38)(24,39)(25,40)(27,44)(28,43)(31,37)(32,42)(35,45)(36,41)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 590);