Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#22694 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | D42 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| socle | C2×C17 | D4×C4⋊C4 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ G | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,24,29,25,30,23,42)(20,31,33,26,35,44,34,45)(21,36,32,46,38,43,37,27)(22,28,39,48,41,49,40,47);(18,20)(19,26)(21,22)(23,34)(24,33)(25,35)(27,28)(29,44)(30,45)(31,42)(32,39)(36,48)(37,40)(38,41)(43,47)(46,49);(18,21,23,37,25,38,24,32)(19,27,29,36,30,46,42,43)(20,22,34,40,35,41,33,39)(26,28,44,48,45,49,31,47);(18,22)(19,28)(20,32)(21,34)(23,40)(24,39)(25,41)(26,43)(27,44)(29,48)(30,49)(31,46)(33,38)(35,37)(36,45)(42,47)986 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15682);