Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C17×#7872 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C17 | D4 |
| commutator | C4 | C2×C4×C8×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| socle | C23×C17 | C2×C4⋊C4 |
| prime radical | C24×C17 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C17 ◁ C22×C42×C17 ◁ G | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 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)(32,40)(33,37)(34,38)(35,39)(36,41)(42,45)(43,44);(19,21)(30,32,34,42,35,43,33,41)(31,36,37,44,39,45,38,40);(30,33,35,34)(31,37,39,38)(32,41,43,42)(36,44,45,40)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 860);