Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #397 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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)(14,22,25,35)(15,19)(16,20)(17,21)(18,24,32,29)(23,38,26,37)(27,33)(28,34)(30,36)(31,40,42,39)(41,43);(12,14)(13,18)(15,26)(16,29)(17,25)(19,31)(20,35)(21,32)(22,36)(23,28)(24,30)(27,39)(33,38)(34,42)(37,43)(40,41);(12,15,16,27,17,28,30,41)(13,19,20,33,21,34,36,43)(14,23,24,39,25,26,29,40)(18,31,22,37,32,42,35,38)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 812);