Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C23×C13 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C42×C13 ◁ G | |
208 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 88 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)(15,20,22,34)(16,25)(17,29)(21,36,37,45)(23,35,24,30)(27,42)(28,41)(31,32)(33,38,44,39)(40,43);(14,15,18,23)(16,21,28,38)(17,30,29,34)(19,24,26,22)(20,32,35,31)(25,39,41,37)(27,36,43,44)(33,42,45,40);(14,16,26,41)(15,21,22,37)(17,27,32,42)(18,28,19,25)(20,33,34,44)(23,38,24,39)(29,43,31,40)(30,36,35,45)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 247);