Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C2×C13×#510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C13 |
| prime radical | C2×C13×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
360 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47);(3,4)(5,13)(6,17)(7,22)(8,10)(9,14)(11,20)(12,19)(15,33)(16,30)(18,28)(21,29)(23,27)(24,32)(25,34)(26,31);(3,5,8,16)(4,9,10,24)(6,18,21,34)(7,23,19,15)(11,27,26,33)(12,28,22,25)(13,29,30,17)(14,31,32,20);(4,10)(9,24)(11,26)(12,22)(13,30)(17,29)(25,28)(27,33)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2145);