Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1384 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #106 |
| commutator | C8 | C42×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| socle | C2×C11 | #340 |
| prime radical | C23×C11 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
132 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)(14,24,26,38)(15,21)(16,31)(17,33)(18,23)(19,25,34,39)(20,29)(22,32)(27,42,40,36)(28,43,41,35)(30,37);(12,14,16,27,32,41,30,39,18,26,29,40,17,28,15,25)(13,19,21,35,33,42,20,24,23,34,37,43,22,36,31,38)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 116);