Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1909 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #386 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
312 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 108 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)(14,22)(15,26,28,19)(16,29)(17,21)(18,24)(20,31)(23,36,39,43)(25,37)(27,34,30,35)(32,38,42,40)(33,41);(12,14,16,24,17,25,31,41)(13,18,20,22,21,33,29,37)(15,23,30,40,28,39,27,38)(19,32,35,36,26,42,34,43);(12,15)(13,19)(14,23)(16,30)(17,28)(18,32)(20,35)(21,26)(22,36)(24,40)(25,39)(27,31)(29,34)(33,42)(37,43)(38,41)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 641);