Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2047 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C23×C11 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
256 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,21,25)(18,26,29,41)(19,30,33,23)(20,34,36,24)(22,35,38,28)(27,43,31,42)(32,40,47,45)(37,44,39,46);(16,18,21,29)(17,22,25,38)(19,31,33,27)(20,35,36,28)(23,39,30,37)(24,41,34,26)(32,44,47,46)(40,42,45,43);(13,15)(16,19,20,32,21,33,36,47)(17,23,24,40,25,30,34,45)(18,27,28,44,29,31,35,46)(22,37,26,42,38,39,41,43)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 779);