Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#11393 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | #120 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C23×C4 | C23×C17 |
| socle | C22×C17 | #379 |
| prime radical | C24×C17 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C23×C4×C17 ◁ G | |
850 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)(26,27,31,35,32,36,30,43)(28,37,39,33,40,48,44,47)(29,34,41,50,42,51,45,49)(38,53,56,57,55,52,54,46);(18,19,21,23)(20,24,25,22)(26,28)(27,33)(29,38)(30,44)(31,39)(32,40)(34,46)(35,48)(36,47)(37,43)(41,56)(42,55)(45,54)(49,52)(50,53)(51,57);(18,20,21,25)(19,22,23,24)(26,29)(27,34)(28,38)(30,45)(31,41)(32,42)(33,46)(35,50)(36,51)(37,52)(39,56)(40,55)(43,49)(44,54)(47,57)(48,53)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 4381);