Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C23×C13 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ G | |
202 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 68 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)(14,15,20,35)(16,25,27,42)(17,30,36,32)(18,23,19,24)(21,37,38,26)(22,34,45,33)(28,41,29,43)(31,39,44,40);(14,16,18,28)(15,21,23,39)(17,31,33,26)(19,29,20,27)(22,41,30,25)(24,40,35,38)(32,42,45,43)(34,37,36,44);(14,15,19,24)(16,21,29,40)(17,32,34,22)(18,23,20,35)(25,26,43,44)(27,38,28,39)(30,33,45,36)(31,42,37,41)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 349);