Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#11194 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | #120 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C23×C4 | C23×C17 |
| socle | C22×C17 | #386 |
| prime radical | C22×#41×C17 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C23×C4×C17 ◁ G | |
822 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)(22,23)(24,34)(25,30,41,49)(26,33)(27,31)(28,32)(29,36)(35,46,52,50)(37,44)(38,45)(39,43,51,53)(40,48,42,47);(18,19)(20,21)(22,24)(23,29)(25,39)(26,38)(27,36)(28,37)(30,46)(31,44)(32,45)(33,34)(35,42)(40,52)(41,51)(43,48)(47,53)(49,50);(19,21)(22,25,26,40,28,41,27,42)(23,30,31,47,32,49,33,48)(24,35,36,51,37,52,38,39)(29,43,34,50,45,53,44,46)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 4182);