Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C11×C16 · Schur multiplier (H2) = C2 |
| direct product | C11×#1401 |
| subgroup | G/H | |
|---|---|---|
| center | C11×C16 | D4 |
| commutator | C4 | C2×C11×C16 |
| Frattini, cosocle | C2×C16 | C22×C11 |
| socle | C2×C11 | C22⋊C16 |
| prime radical | C11×#41 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C11×C16 ◁ C2×C11×C16 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 75 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,22,17,23,35,48,18,24,36,49,38,51,63,71,19,25,37,50,39,52,64,72,15,21,32,45,33,46,60,69)(14,26,28,53,29,54,57,73,27,43,55,47,56,67,74,70,30,44,34,61,58,68,62,75,31,20,40,41,59,42,65,66);(12,14)(13,20)(15,31)(16,34)(17,29)(18,27)(19,30)(21,44)(22,47)(23,42)(24,26)(25,43)(28,37)(32,55)(33,59)(35,62)(36,40)(38,56)(39,58)(41,50)(45,53)(46,68)(48,70)(49,61)(51,54)(52,67)(57,64)(60,74)(63,65)(66,72)(69,73)(71,75)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 133);