Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C17×#22616 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C17 | C22×D4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C22×C4 | C24×C17 |
| prime radical | C24×C17 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C4×C22⋊C4×C17 ◁ G | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2176 |
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)(22,23)(24,26)(25,27)(28,29)(30,31,35,34,36,39,33,38)(32,40,41,42,43,44,45,37);(22,24,25,28)(23,26,27,29)(30,32)(31,37)(33,41)(34,44)(35,45)(36,43)(38,40)(39,42);(23,27)(26,29)(30,33,36,35)(31,38,39,34)(32,41,43,45)(37,40,42,44);(30,34,35,39,36,38,33,31)(32,42,41,44,43,37,45,40)986 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 15604);