Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#3022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C11×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C4×C11 ◁ G | |
704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 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 — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,18,23)(14,24,27,36)(15,20,29,35)(16,22,17,21)(19,30,33,26)(25,39,42,38)(28,37,31,34)(32,40,43,41);(12,14)(13,19)(15,25)(16,26)(17,30)(18,27)(20,32)(21,24)(22,36)(23,33)(28,40)(29,42)(31,41)(34,38)(35,43)(37,39);(12,15)(13,20)(14,25)(16,28)(17,31)(18,29)(19,32)(21,34)(22,37)(23,35)(24,38)(26,40)(27,42)(30,41)(33,43)(36,39);(12,16,18,17)(13,21,23,22)(14,26,27,30)(15,28,29,31)(19,24,33,36)(20,34,35,37)(25,40,42,41)(32,38,43,39)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1754);