Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | D42 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C22×C4×C17 ◁ G | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 4 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)(24,36)(25,40)(26,32)(27,34)(28,33)(29,35)(30,38)(31,41)(37,51)(39,49)(42,50)(43,53)(44,47)(45,48)(46,52);(22,24,29,39)(23,30,35,48)(25,26,42,44)(27,45,28,38)(31,47,51,32)(33,49,34,36)(37,46,41,43)(40,53,50,52);(22,25,28,41,29,42,27,37)(23,31,34,40,35,51,33,50)(24,26,38,43,39,44,45,46)(30,47,36,53,48,32,49,52);(22,26)(23,32)(24,37)(25,38)(27,46)(28,43)(29,44)(30,40)(31,49)(33,53)(34,52)(35,47)(36,51)(39,41)(42,45)(48,50)986 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15678);