Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#7721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| prime radical | C24×C17 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C22×C42×C17 ◁ G | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,24,25,22)(21,23)(26,27)(28,35,37,40)(29,38)(30,33)(31,34)(32,36,41,39)(42,43,45,48,46,49,52,55)(44,47,51,54,50,53,56,57);(18,20)(19,22)(21,25)(23,24)(26,28,31,37)(27,32,34,41)(29,36,30,39)(33,40,38,35)(42,44)(43,47)(45,51)(46,50)(48,54)(49,53)(52,56)(55,57);(19,23)(22,24)(26,29,31,30)(27,33,34,38)(28,36,37,39)(32,40,41,35)(44,50)(47,53)(51,56)(54,57)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 709);