Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C11×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
208 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 64 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)(12,13,19,29)(14,23,34,43)(15,17,22,18)(16,28,35,39)(20,36,42,25)(21,32,31,33)(24,40,37,30)(26,38,27,41);(12,14)(13,20)(15,24)(16,30)(17,27)(18,26)(19,34)(21,38)(22,37)(23,28)(25,33)(29,42)(31,41)(32,36)(35,40)(39,43);(12,15,18,13)(14,24,26,20)(16,31,33,39)(17,29,19,22)(21,32,28,35)(23,40,38,36)(25,43,30,41)(27,42,34,37)253 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 347);