Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1542 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C22⋊C4 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C42 | C23×C13 |
| socle | C22×C13 | C2×C22⋊C4 |
| prime radical | C23×C13 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
212 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,21,19,16,17,20)(22,23,26,31,27,30,25,29)(24,32,33,37,34,28,35,36);(14,16,19,15)(17,20,18,21)(22,24)(23,28)(25,33)(26,35)(27,34)(29,36)(30,32)(31,37);(22,25,27,26)(23,29,30,31)(24,33,34,35)(28,36,32,37)494 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 274);