Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#12997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | #447 |
| 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 | |
492 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)(19,21)(22,23,27,34,26,33,42,30,29,36,25,32,28,35,43,46)(24,37,40,45,44,31,47,49,41,51,38,52,39,50,53,48);(18,19,20,21)(23,30)(25,42)(26,28)(27,43)(31,45)(32,35)(33,34)(36,46)(37,49)(38,47)(39,44)(40,53)(48,51)(50,52);(22,24)(23,31)(25,40)(26,44)(27,38)(28,39)(29,41)(30,45)(32,49)(33,51)(34,48)(35,37)(36,50)(42,53)(43,47)(46,52)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5985);