Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1407 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #353 |
| commutator | C42 | C2×C4×C13 |
| Frattini, cosocle | #124 | C22×C13 |
| prime radical | C2×D4×C13 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C13 ◁ C2×C4×C13 ◁ C13×#124 ◁ G | |
98 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)(16,24,26,39)(17,28,29,44)(18,31,33,22)(19,23)(20,25,34,41)(21,30)(27,37,32,40)(35,42,36,43)(38,45);(14,16,19,27)(15,20,23,36)(17,25,29,43)(18,32,33,26)(21,37,38,24)(22,34,31,35)(28,40,44,39)(30,41,45,42)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 139);