Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#2889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C2×C11×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C11×#41 ◁ G | |
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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 — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,21,17,20,29,34)(14,22,24,35,25,18,28,31)(15,19,27,33,26,32,39,42)(23,36,38,43,37,30,40,41);(12,14,17,25)(13,18,20,22)(15,23,26,37)(16,28,29,24)(19,30,32,36)(21,35,34,31)(27,40,39,38)(33,43,42,41);(12,15)(13,19)(14,23)(16,27)(17,26)(18,30)(20,32)(21,33)(22,36)(24,38)(25,37)(28,40)(29,39)(31,41)(34,42)(35,43);(15,26)(19,32)(23,37)(27,39)(30,36)(33,42)(38,40)(41,43)451 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1621);