Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#22731 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C17 | C22×D4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C2×C4×Q8×C17 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 4352 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21)(23,26)(27,28)(30,31)(32,39)(33,37)(34,43)(35,38)(36,42)(40,45)(41,44);(22,23,25,26)(24,28,29,27)(32,40)(36,44)(39,45)(41,42);(30,32,33,41,35,40,34,42)(31,36,43,45,38,44,37,39);(22,24,25,29)(23,27,26,28)(30,33,35,34)(31,37,38,43)(32,41,40,42)(36,39,44,45)986 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15719);