Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#1289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #95 |
| commutator | C2×C4 | C42×C13 |
| Frattini, cosocle | C23×C4 | C22×C13 |
| prime radical | C24×C13 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C23×C4×C13 ◁ G | |
228 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 416 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,19)(16,17)(20,21)(22,23)(24,28,29,26)(25,27);(14,15,17,20,18,21,16,19)(22,24)(23,26)(25,29)(27,28)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 21);