Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2117 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C2×D4 |
| commutator | C4 | C22×C8×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| socle | C2×C13 | C2×C22⋊C8 |
| prime radical | C2×C13×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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)(14,15,18,24,19,25,33,42,20,22,34,38,17,23,31,40)(16,26,28,43,27,36,44,41,29,39,32,45,30,21,35,37);(14,16)(15,21)(17,30)(18,32)(19,27)(20,29)(22,36)(23,39)(24,41)(25,26)(28,34)(31,44)(33,35)(37,38)(40,43)(42,45);(15,22)(21,36)(23,25)(24,38)(26,39)(37,41)(40,42)(43,45)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 849);