Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1411 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #353 |
| commutator | C42 | C2×C4×C13 |
| Frattini, cosocle | C4⋊Q8 | C22×C13 |
| prime radical | C22×C13 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C13 ◁ C2×C4×C13 ◁ C13×C4⋊Q8 ◁ G | |
94 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)(14,15,17,22,19,24,31,41)(16,25,26,39,28,36,43,44)(18,23,33,40,34,21,30,38)(20,29,35,32,37,42,45,27);(14,16,19,28)(15,20,24,37)(17,29,31,42)(18,32,34,27)(21,36,23,25)(22,39,41,44)(26,30,43,33)(35,40,45,38)221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 143);