Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1389 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #102 |
| commutator | C8 | C42×C13 |
| Frattini, cosocle | C22×C8 | C22×C13 |
| socle | C2×C13 | #340 |
| prime radical | C23×C13 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
128 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,23,20,24,19,35)(16,25,27,42,30,38,29,44)(17,31,32,40,34,45,33,22)(21,36,37,28,39,43,41,26);(14,16,17,28,18,27,32,43,20,30,34,26,19,29,33,36)(15,21,22,38,35,41,45,42,24,39,40,25,23,37,31,44)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 121);