Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C4×C13 |
| Frattini, cosocle | #124 | C22×C13 |
| prime radical | C13×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C22×C13 ◁ C2×C4×C13 ◁ C13×#124 ◁ G | |
112 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,20)(17,26,27,23)(18,22,25,29)(19,28,21,24);(14,15)(16,23)(17,26)(18,22)(19,25)(20,24)(21,28)(27,29)221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 136);