Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1379 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | D4 |
| commutator | C4 | C4×C8×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C22×C13 | C22⋊C8 |
| prime radical | C23×C13 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C22×C8×C13 ◁ G | |
136 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 136 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) = 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)(18,19,22,27,23,28,37,45,24,29,38,46,21,26,35,43)(20,30,32,47,31,41,48,44,33,42,36,49,34,25,39,40);(14,15,16,17)(18,20)(19,25)(21,34)(22,36)(23,31)(24,33)(26,42)(27,44)(28,30)(29,41)(32,38)(35,48)(37,39)(40,46)(43,47)(45,49)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 111);