Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
342 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 92 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)(16,17,22,27)(18,28,32,39)(19,33,37,42)(20,26,21,25)(23,31,40,30)(24,36,43,35)(29,44,46,38)(34,41,45,47);(13,15)(16,18)(17,23)(19,34)(20,31)(21,30)(22,32)(24,41)(25,39)(26,28)(27,40)(29,36)(33,44)(35,46)(37,45)(38,42)(43,47);(16,19,22,37)(17,24,27,43)(18,29,32,46)(20,36,21,35)(23,38,40,44)(25,42,26,33)(28,41,39,47)(30,34,31,45)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 752);