Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2017 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
350 conjugacy classes of subgroups.
| 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)(14,23)(15,28)(16,33)(17,21)(18,22)(19,25)(20,31)(24,41)(26,36)(27,35)(29,40)(30,39)(32,38)(34,42)(37,43);(12,14)(13,19)(15,29)(16,26)(17,25)(18,27)(20,37)(21,35)(22,36)(23,33)(24,30)(28,34)(31,42)(32,43)(38,41)(39,40);(12,15)(13,20)(14,24)(16,30)(17,31)(18,32)(19,34)(21,38)(22,39)(23,40)(25,29)(26,43)(27,42)(28,33)(35,37)(36,41)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 749);