Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #120 |
| commutator | C23 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | #379 |
| prime radical | C13×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,23,19,22,31,36)(16,24,27,37,26,20,30,33)(17,21,29,34,28,35,41,45)(25,32,40,43,39,38,42,44);(14,16)(15,20)(17,25)(18,30)(19,26)(21,32)(22,24)(23,37)(27,31)(28,39)(29,42)(33,36)(34,44)(35,38)(40,41)(43,45);(14,17)(15,21)(16,25)(18,29)(19,28)(20,32)(22,35)(23,34)(24,38)(26,39)(27,40)(30,42)(31,41)(33,43)(36,45)(37,44)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 523);