Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#2949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator | C4 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C22×D4×C13 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(18,19)(20,26)(21,29)(22,24)(23,27)(25,31)(28,33)(30,32);(18,20,22,28)(19,23,24,32)(21,30,31,27)(25,33,29,26);(14,16,17,15)(18,19)(20,23)(21,25)(22,24)(26,27)(28,32)(29,31)(30,33);(19,24)(23,32)(25,29)(26,33)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1681);