Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#2242 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C2×C11×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ C4×C8×C11 ◁ G | |
138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21)(22,34)(23,28)(24,42)(25,45)(26,46)(27,33)(29,38)(30,35)(31,44)(32,43)(36,48)(37,47)(39,51)(40,50)(41,49);(13,16)(17,19)(20,22,35,40,26,37,44,41,27,38,25,36,43,39,24,23)(21,28,42,51,32,48,45,29,33,49,31,47,46,50,30,34);(12,14,15,18)(13,17,16,19)(20,23,24,39,43,36,25,38,27,41,44,37,26,40,35,22)(21,29,30,48,46,51,31,28,33,34,45,50,32,47,42,49)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 974);