Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#1287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #95 |
| commutator | C2×C4 | C42×C13 |
| Frattini, cosocle | C2×C42 | C22×C13 |
| prime radical | C23×C13 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C2×C42×C13 ◁ G | |
126 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 832 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,22,19,23,18,21)(16,24,25,20,27,28,26,29)(30,31)(32,38,40,44)(33,36)(34,37)(35,39,42,45)(41,43);(14,16,18,26,19,27,17,25)(15,20,22,28,23,29,21,24)(30,32)(31,35,37,42)(33,39)(34,40)(36,38,43,44)(41,45)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 19);