Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1378 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C11 | C22⋊C4 |
| commutator | C22 | C4×C8×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C2×C11 | #336 |
| prime radical | C23×C11 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C11 ◁ C22×C8×C11 ◁ G | |
132 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 84 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 — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,19,16,21,26,33,17,20,27,32,28,34,39,42)(14,22,24,35,25,36,38,43,23,18,29,30,37,31,40,41);(12,14)(13,18,20,22)(15,24)(16,25)(17,23)(19,30,32,35)(21,31,34,36)(26,38)(27,29)(28,37)(33,41,42,43)(39,40)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 110);