Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#3067 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C2×C11 | C2×#121 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
452 conjugacy classes of subgroups.
| 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)(14,24)(15,29)(16,21)(17,22)(18,23)(19,26)(20,30)(25,39)(27,35)(28,34)(31,36)(32,37)(33,41)(38,40)(42,43);(12,14,16,26,18,28,17,27)(13,19,22,24,23,35,21,34)(15,25,30,40,31,41,32,42)(20,33,29,38,37,39,36,43);(12,15)(13,20)(14,25)(16,30)(17,32)(18,31)(19,33)(21,36)(22,29)(23,37)(24,38)(26,40)(27,42)(28,41)(34,43)(35,39);(12,16,18,17)(13,21,23,22)(14,26,28,27)(15,30,31,32)(19,34,35,24)(20,36,37,29)(25,40,41,42)(33,43,39,38)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1799);