Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C13×C16 · Schur multiplier (H2) = C2 |
| direct product | C13×#1400 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C22⋊C4 |
| commutator | C22 | C2×C13×C16 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| socle | C2×C13 | C22⋊C16 |
| prime radical | C23×C13 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C2×C13×C16 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,21,17,22,25,33,18,23,26,34,27,35,38,43,19,20,24,30,28,31,36,40,29,32,37,41,39,42,44,45);(15,20)(16,24)(22,31)(23,32)(25,36)(26,37)(35,42)(38,44)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 132);