Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C2×#99 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C2×Q8 | C23×C13 |
| prime radical | C2×C13×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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)(14,15,19,24,20,25,17,22)(16,26,29,44,30,41,27,43)(18,32,31,42,34,23,35,40)(21,36,37,33,38,45,39,28);(14,16,20,30)(15,21,25,38)(17,29,19,27)(18,33,34,28)(22,39,24,37)(23,41,32,26)(31,36,35,45)(40,43,42,44);(14,17,20,19)(15,22,25,24)(16,27,30,29)(18,31,34,35)(21,37,38,39)(23,40,32,42)(26,43,41,44)(28,36,33,45)299 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 867);