Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | D8 |
| commutator | C8 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | C4×D8 |
| prime radical | C11×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ G | |
146 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 76 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)(12,13,17,21,18,23,33,22)(14,24,26,35,28,41,27,43)(15,20,29,38,30,39,16,31)(19,34,36,42,37,32,40,25);(12,14)(13,19)(15,25)(16,32)(17,26)(18,28)(20,35)(21,36)(22,40)(23,37)(24,31)(27,33)(29,34)(30,42)(38,41)(39,43);(12,15,17,29,18,30,33,16)(13,20,21,38,23,39,22,31)(14,25,26,34,28,42,27,32)(19,35,36,41,37,43,40,24)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 910);