Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1357 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1358 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #331 |
| commutator | C42 | C2×C4×C13 |
| Frattini, cosocle | C4×C8 | C22×C13 |
| prime radical | C22×C13 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C4×C8×C13 ◁ G | |
92 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)(14,15,17,22,18,21,30,37,19,24,31,40,33,23,41,39)(16,25,26,42,27,36,43,45,28,35,44,38,32,20,29,34);(14,16,19,28)(15,20,24,36)(17,29,31,43)(18,32,33,27)(21,35,23,25)(22,38,40,42)(26,30,44,41)(34,37,45,39)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 90);