Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C2×C13 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,26)(17,31)(18,23)(19,24)(20,25)(21,28)(22,32)(27,41)(29,37)(30,36)(33,38)(34,39)(35,43)(40,42)(44,45);(14,16,18,28,20,30,19,29)(15,21,24,26,25,37,23,36)(17,27,32,42,33,43,34,44)(22,35,31,40,39,41,38,45);(14,17)(15,22)(16,27)(18,32)(19,34)(20,33)(21,35)(23,38)(24,31)(25,39)(26,40)(28,42)(29,44)(30,43)(36,45)(37,41);(14,18,20,19)(15,23,25,24)(16,28,30,29)(17,32,33,34)(21,36,37,26)(22,38,39,31)(27,42,43,44)(35,45,41,40)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1799);