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) = C25 |
| direct product | C2×C13×#342 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22⋊C4 |
| commutator | C22 | C2×C42×C13 |
| Frattini, cosocle | C24 | C23×C13 |
| prime radical | C25×C13 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C25×C13 ◁ G | |
680 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27);(7,8)(9,13,14,11)(10,12);(3,4,5,6)(7,9)(8,11)(10,14)(12,13)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 471);