Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#2251 |
| 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 | |
178 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 178 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)(16,17,21,25)(18,26,28,40)(19,31,33,46)(20,34,32,24)(22,35,36,29)(23,38,39,27)(30,45,44,42)(37,43,41,47);(16,18,27,43,20,30,33,35,21,28,38,47,32,44,19,29)(17,22,31,42,24,37,39,40,25,36,46,45,34,41,23,26);(12,13,14,15)(18,28)(22,36)(26,40)(29,35)(30,44)(37,41)(42,45)(43,47)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 983);