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) = C2 |
| direct product | C11×#1417 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | D16 |
| commutator | C16 | C2×C4×C11 |
| Frattini, cosocle | C2×C16 | C22×C11 |
| socle | C2×C11 | #357 |
| prime radical | C11×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ 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)(12,13,16,21,18,24,37,23)(14,25,27,54,30,55,29,61)(15,31,34,62,36,64,69,63)(17,38,33,22,40,50,68,52)(19,41,42,57,44,56,53,26)(20,45,47,70,49,72,48,35)(28,59,58,73,60,74,39,46)(32,51,65,71,67,75,66,43);(12,14)(13,19)(15,32)(16,27)(17,39)(18,30)(20,46)(21,42)(22,51)(23,53)(24,44)(25,31)(26,35)(28,33)(29,37)(34,65)(36,67)(38,43)(40,58)(41,45)(47,59)(48,74)(49,73)(50,71)(52,75)(54,62)(55,64)(56,72)(57,70)(60,68)(61,63)(66,69)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 149);