Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1382 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | D8 |
| commutator | C8 | C42×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C22×C13 | #102 |
| prime radical | C23×C13 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
132 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 2 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,29)(21,34)(22,37)(23,39)(24,28)(25,30)(26,36)(27,35)(31,41)(32,49)(33,45)(38,42)(40,48)(43,47)(44,46);(18,20,22,31,35,46,36,47,24,33,21,30,23,32,38,48)(19,25,26,29,39,43,37,49,28,41,42,45,27,40,34,44)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 114);