Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2094 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C2×C42×C11 ◁ G | |
290 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 290 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) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(16,17,19,21)(18,22,23,20)(24,25,28,29,27,31,26,30);(12,13)(14,15)(16,18,19,23)(17,20,21,22)(25,29)(26,28)(30,31);(13,15)(25,29)(26,28)(30,31)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 826);