Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#2614 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
782 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 782 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) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(17,19)(20,21,24,27)(22,28,30,25)(23,26,32,31)(29,34,35,33);(13,15)(17,19)(20,22,24,30)(21,25,27,28)(23,29,32,35)(26,33,31,34);(12,13)(14,15)(16,17)(18,19)(20,21)(22,28)(23,31)(24,27)(25,30)(26,32)(29,33)(34,35);(20,23)(21,26)(22,29)(24,32)(25,33)(27,31)(28,34)(30,35)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1346);