Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#1757 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C22⋊C4 |
| commutator | C22 | C2×C42×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C2×C13 | C4×C22⋊C4 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ G | |
284 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 148 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)(14,15,18,22,19,23,30,36)(16,24,27,38,26,20,31,33)(17,21,28,34,29,35,41,44)(25,37,40,45,39,32,42,43);(14,16)(15,20,23,24)(17,25)(18,27)(19,26)(21,32,35,37)(22,33,36,38)(28,40)(29,39)(30,31)(34,43,44,45)(41,42);(14,17,18,28,19,29,30,41)(15,21,22,34,23,35,36,44)(16,25,27,40,26,39,31,42)(20,32,33,43,24,37,38,45)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 489);