Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#1537 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C42 | C23×C11 |
| prime radical | C23×C11 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C42×C11 ◁ G | |
194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,19,16,18,24,26)(14,20,21,27,22,17,23,25)(28,29)(30,37,39,42)(31,40)(32,35)(33,36)(34,38,43,41);(12,14,16,22)(13,17,18,20)(15,23,24,21)(19,27,26,25)(28,30)(29,34)(31,38)(32,41)(33,39)(35,42)(36,43)(37,40);(28,31,33,32)(29,35,36,40)(30,38,39,41)(34,42,43,37)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 269);