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) = C2 |
| direct product | C13×#1417 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | D16 |
| commutator | C16 | C2×C4×C13 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle | C2×C13 | #357 |
| prime radical | C13×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C4×C13 ◁ C2×C8×C13 ◁ C2×C13×C16 ◁ G | |
72 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)(14,15,18,23,20,26,39,25)(16,27,29,56,32,57,31,63)(17,33,36,64,38,66,71,65)(19,40,35,24,42,52,70,54)(21,43,44,59,46,58,55,28)(22,47,49,72,51,74,50,37)(30,61,60,75,62,76,41,48)(34,53,67,73,69,77,68,45);(14,16)(15,21)(17,34)(18,29)(19,41)(20,32)(22,48)(23,44)(24,53)(25,55)(26,46)(27,33)(28,37)(30,35)(31,39)(36,67)(38,69)(40,45)(42,60)(43,47)(49,61)(50,76)(51,75)(52,73)(54,77)(56,64)(57,66)(58,74)(59,72)(62,70)(63,65)(68,71)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 149);