Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2115 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C22⋊C4 |
| commutator | C22 | C22×C8×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C2×C22⋊C8 |
| prime radical | C2×C11×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C22×C8×C11 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 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)(12,13,15,20,16,21,27,35,17,19,28,32,29,33,40,42)(14,22,24,34,25,36,38,43,23,18,26,30,37,31,39,41);(12,14)(13,18)(15,26)(16,25)(17,23)(19,22)(20,34)(21,31)(24,28)(27,39)(29,37)(30,32)(33,36)(35,43)(38,40)(41,42);(14,23)(18,22)(24,26)(25,37)(30,34)(31,36)(38,39)(41,43)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 847);