Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#2070 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C23×C13 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C4×C13 ◁ G | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,32,19,23,34,22)(16,24,26,20,27,40,44,36)(17,28,30,37,31,21,33,38)(25,41,42,35,29,39,43,45);(14,16,18,26,19,27,34,44)(15,20,22,24,23,36,32,40)(17,29,30,43,31,25,33,42)(21,35,37,41,28,45,38,39);(14,17,19,31)(15,21,23,28)(16,25,27,29)(18,33,34,30)(20,35,36,45)(22,38,32,37)(24,39,40,41)(26,43,44,42)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 802);