Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C23×C11 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×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) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,22,18,23,16,21)(14,24,27,19,28,35,26,36)(15,29,31,37,32,38,33,20)(25,40,41,43,30,34,42,39);(12,14,18,28)(13,19,23,36)(15,30,32,25)(16,27,17,26)(20,34,37,40)(21,35,22,24)(29,43,38,39)(31,41,33,42);(12,15,16,31,18,32,17,33)(13,20,21,29,23,37,22,38)(14,25,26,41,28,30,27,42)(19,34,24,39,36,40,35,43)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 814);