Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1413 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #354 |
| commutator | C42 | C2×C4×C11 |
| Frattini, cosocle | C4⋊Q8 | C22×C11 |
| prime radical | C22×C11 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×C4×C11 ◁ C11×C4⋊Q8 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,20,17,22,29,39)(14,23,25,41,24,40,42,37)(16,21,31,38,32,19,28,36)(18,33,34,26,35,27,43,30);(12,14,17,24)(13,18)(15,27,29,33)(16,30,32,26)(19,23)(20,37)(21,40)(22,35)(25,28,42,31)(34,36)(38,43)(39,41)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 145);