Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C1 |
| direct product | C11×#1425 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | D16 |
| commutator | C16 | C2×C4×C11 |
| Frattini, cosocle | C2×C16 | C22×C11 |
| socle | C2×C11 | #366 |
| prime radical | C22×C11 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ G | |
50 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)(12,13,16,22,19,24,18,40)(14,25,28,50,31,52,30,62)(15,32,34,63,36,65,35,66)(17,38,37,48,39,67,33,23)(20,41,42,58,44,57,68,27)(21,45,46,55,47,54,64,26)(29,60,59,72,61,75,56,51)(43,53,49,69,70,74,71,73);(12,14,26,53,17,29,34,41,16,28,45,69,37,59,36,58,19,31,55,74,39,61,35,57,18,30,54,73,33,56,15,27)(13,20,32,51,23,43,64,62,40,68,66,75,67,71,47,52,24,44,65,72,48,70,46,50,22,42,63,60,38,49,21,25)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 157);