Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C26 |
| direct product | C13×#3025 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C13 |
| socle | C2×C13 | C2×#120 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ G | |
766 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 212 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)(19,25)(26,29);(14,15)(16,19)(17,20)(18,21)(22,26)(23,25)(24,27)(28,29);(14,16)(15,19)(17,22)(18,23)(20,26)(21,25)(24,28)(27,29);(14,17,18,24)(15,20,21,27)(16,22,23,28)(19,26,25,29)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1757);