Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1354 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #99 |
| commutator | C22×C4 | C2×C4×C13 |
| Frattini, cosocle | C2×C4⋊C4 | C22×C13 |
| prime radical | C23×C13 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C23×C13 ◁ C2×C13×C4⋊C4 ◁ G | |
140 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 40 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)(15,18,19,20)(22,23,25,29,26,28,24,27);(14,15)(16,20)(17,19)(18,21)(22,24,26,25)(23,27)(28,29)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 86);