Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #354 |
| 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 | |
78 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)(16,25,27,43,26,42,44,39)(18,23,33,40,34,21,30,38)(20,35,36,28,37,29,45,32);(14,16,19,26)(15,20)(17,29,31,35)(18,32,34,28)(21,25)(22,39)(23,42)(24,37)(27,30,44,33)(36,38)(40,45)(41,43)182 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 145);