Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1394 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #99 |
| commutator | C2×C4 | C42×C13 |
| Frattini, cosocle | C22×Q8 | C22×C13 |
| prime radical | C23×C13 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C23×C13 ◁ C22×Q8×C13 ◁ G | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,21,25,22,26,30,33)(20,24,27,32,28,23,29,31);(14,15,16,17)(19,23,26,24)(25,32,33,31)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 126);