Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1554 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C42 | C23×C13 |
| prime radical | C23×C13 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C42×C13 ◁ G | |
198 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 198 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,21,19,22,17,26)(16,23,24,28,25,20,27,29)(30,31)(32,38)(33,41)(34,36)(35,39)(37,43)(40,45)(42,44);(14,16,19,25)(15,20,22,23)(17,24,18,27)(21,28,26,29)(30,32,34,40)(31,35,36,44)(33,42,43,39)(37,45,41,38);(14,17,19,18)(15,21,22,26)(16,24,25,27)(20,28,23,29)(31,36)(35,44)(37,41)(38,45)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 286);