Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#12842 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #466 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C2×#41×C17 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C22×C4×C17 ◁ C2×C17×#131 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 2176 |
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)(22,23,27,31,28,25,39,32)(24,33,37,48,38,50,53,49)(26,41,34,30,42,46,51,40)(29,43,44,47,35,52,45,36);(22,24,34,43,27,37,42,47,28,38,51,52,39,53,26,36)(23,29,41,49,32,45,40,50,25,35,46,48,31,44,30,33);(19,21)(22,25)(23,28)(24,35)(26,40)(27,31)(29,38)(30,42)(32,39)(33,47)(34,46)(36,50)(37,44)(41,51)(43,48)(45,53)(49,52)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5830);