Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#8409 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | #120 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| socle | C22×C17 | #379 |
| prime radical | C17×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C42×C17 ◁ G | |
972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1088 |
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)(26,27,30,35,31,34,45,49)(28,36,39,50,38,32,43,47)(29,40,42,51,41,33,44,48)(37,52,54,57,53,46,55,56);(19,23)(20,21)(24,25)(26,28)(27,32)(29,37)(30,43)(31,38)(33,46)(34,36)(35,50)(39,45)(40,52)(41,53)(42,55)(44,54)(47,49)(48,57)(51,56);(26,29)(27,33)(28,37)(30,44)(31,41)(32,46)(34,40)(35,51)(36,52)(38,53)(39,55)(42,45)(43,54)(47,57)(48,49)(50,56)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 1397);