Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1414 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #354 |
| 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 | |
84 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)(17,28,30,44)(18,32,34,22)(19,23)(20,25,35,42)(21,31)(27,41,33,40)(29,36,43,37)(38,45);(14,16)(15,20,23,37)(17,29)(18,33)(19,27)(21,24,38,41)(22,35,32,36)(25,30)(26,34)(28,39,44,40)(31,43)(42,45)221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 146);