Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3172 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C22×C11 | C22×D4 |
| prime radical | C11×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ C2×C42×C11 ◁ G | |
364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,24,27)(22,28,29,35)(23,31,32,26)(25,33,34,30);(13,16)(17,19)(22,29)(25,34)(28,35)(30,33);(20,22,23,30,24,29,32,33)(21,25,26,28,27,34,31,35);(12,14,15,18)(13,17,16,19)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1904);