Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C26 |
| direct product | C13×#2888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C2×C13×H(5,𝔽2) | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C13×#41 ◁ G | |
756 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 340 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,21,18,19,24,26)(16,20,23,27,22,25,28,29);(15,19)(17,24)(20,25)(23,28);(14,16)(15,20)(17,23)(18,22)(19,25)(21,27)(24,28)(26,29);(16,22)(20,25)(23,28)(27,29)533 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1620);