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) = C2×C4 |
| direct product | C17×#7726 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| prime radical | C22×D4×C17 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C43×C17 ◁ G | |
400 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)(18,19,21,23,22,25,20,24)(26,27)(28,34,36,31)(29,37)(30,33)(32,38)(35,39,41,40)(42,43)(44,50,52,57)(45,49)(46,48)(47,51,55,53)(54,56);(19,23)(20,21)(24,25)(26,28)(27,31)(29,35)(30,36)(32,39)(33,34)(37,40)(38,41)(42,44)(43,47)(45,53)(46,52)(48,55)(49,57)(50,56)(51,54);(18,20,22,21)(19,23,25,24)(26,29,30,38)(27,32,33,37)(28,35,36,41)(31,39,34,40)(43,48)(47,55)(49,56)(50,57)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 714);