Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×#1425 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | D16 |
| commutator | C16 | C2×C4×C13 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle | C2×C13 | #366 |
| prime radical | C22×C13 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C8×C13 ◁ C2×C13×C16 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,24,21,26,20,42)(16,27,30,52,33,54,32,64)(17,34,36,65,38,67,37,68)(19,40,39,50,41,69,35,25)(22,43,44,60,46,59,70,29)(23,47,48,57,49,56,66,28)(31,62,61,74,63,77,58,53)(45,55,51,71,72,76,73,75);(14,16,28,55,19,31,36,43,18,30,47,71,39,61,38,60,21,33,57,76,41,63,37,59,20,32,56,75,35,58,17,29)(15,22,34,53,25,45,66,64,42,70,68,77,69,73,49,54,26,46,67,74,50,72,48,52,24,44,65,62,40,51,23,27)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 157);