Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C13 · Schur multiplier (H2) = C214 |
| direct product | C13×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C26 |
| commutator, Frattini, cosocle | C2 | C26×C13 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ G | |
5796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 6 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,21)(17,22)(18,30)(19,20)(23,31)(24,34)(25,39)(26,32)(27,36)(28,42)(29,33)(35,40)(37,43)(38,45)(41,44);(15,20)(21,32)(22,33)(23,30)(34,36)(35,39)(37,42)(44,45);(14,16,19,26)(15,21,20,32)(17,27,29,24)(18,25,31,40)(22,36,33,34)(23,35,30,39)(28,41,43,38)(37,45,42,44);(14,17,19,29)(15,22,20,33)(16,24,26,27)(18,28,31,43)(21,34,32,36)(23,37,30,42)(25,38,40,41)(35,44,39,45);(14,15)(16,21)(17,22)(18,23)(19,20)(24,34)(25,35)(26,32)(27,36)(28,37)(29,33)(30,31)(38,44)(39,40)(41,45)(42,43);(14,18)(15,23)(16,25)(17,28)(19,31)(20,30)(21,35)(22,37)(24,38)(26,40)(27,41)(29,43)(32,39)(33,42)(34,44)(36,45)845 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2327);