Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C23×C11 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
230 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,18)(14,19,15,17)(20,21,23,27)(22,29,30,25)(24,26,32,28)(31,35,33,34);(12,14,16,15)(13,17,18,19)(20,22,23,30)(21,25,27,29)(24,33,32,31)(26,35,28,34);(12,15,16,14)(13,17,18,19)(21,26)(24,32)(25,34)(27,28)(29,35)(31,33)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 793);