Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#2249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C2×C11×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ C4×C8×C11 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 72 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)(13,15)(16,17)(18,26)(19,31)(20,23)(21,34)(22,28)(24,27)(25,32)(29,38)(30,45)(33,46)(35,44)(36,47)(37,42)(39,41)(40,43);(16,18,27,41,21,30,33,44,20,29,42,47,32,43,19,28)(17,22,31,40,25,36,37,38,23,35,46,45,34,39,24,26);(12,13,14,15)(17,23)(22,35)(24,37)(25,34)(26,38)(31,46)(36,39)(40,45)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 981);