Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C13 · Schur multiplier (H2) = C29 |
| direct product | C13×#3428 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C24 |
| commutator | C2 | C24×C4×C13 |
| Frattini, cosocle | C22 | C25×C13 |
| socle | C22×C13 | C25 |
| prime radical | C22×C13×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ G | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,26)(21,25)(22,24)(23,27)(28,32)(29,31)(30,33);(18,20)(19,23)(21,28)(22,27)(24,26)(25,30)(29,33)(31,32);(20,27)(23,26)(28,33)(30,32);(19,24)(23,26)(25,31)(30,32);(18,21,22,29)(19,25,24,31)(20,28,27,33)(23,30,26,32)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2160);