Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #106 |
| commutator | C8 | C42×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| prime radical | C23×C13 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
136 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 80 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)(14,15)(16,17)(19,24,25,26)(20,28)(21,30)(22,29)(27,33,31,32);(15,17)(18,19,21,26,22,27,30,33,23,25,20,24,29,31,28,32)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 115);