Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#12648 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #442 |
| commutator | C8 | C2×C42×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C23×C17 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C2×C17×#97 ◁ G | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,26,31)(24,34)(25,40,27,43)(28,45,41,33)(29,37)(30,44,32,42)(35,50)(36,49)(38,53)(39,47)(46,51)(48,52);(19,21)(22,24,27,37,28,38,44,51,26,39,25,36,41,35,42,52)(23,29,32,47,33,48,43,53,31,49,30,34,45,46,40,50);(24,35,39,38)(25,27)(29,46,49,48)(30,32)(34,50,47,53)(36,52,37,51)(40,43)(42,44)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5636);