Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#7472 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | D16 |
| commutator | C16 | C42×C17 |
| Frattini, cosocle | C22×C16 | C22×C17 |
| socle | C22×C17 | #366 |
| prime radical | C23×C17 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ C22×C16×C17 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,26,33,30,36,29,54)(24,37,39,67,43,68,42,78)(25,34,45,62,47,64,46,79)(27,32,48,60,51,61,50,63)(28,52,49,65,53,80,44,35)(31,55,56,71,59,70,81,38)(40,66,72,83,75,84,74,58)(41,76,73,82,77,85,69,57);(19,21)(22,24,27,40,28,41,45,55,26,39,48,72,49,73,47,71,30,43,51,75,53,77,46,70,29,42,50,74,44,69,25,38)(23,31,34,57,35,58,63,78,54,81,79,85,80,84,61,68,36,59,64,82,65,83,60,67,33,56,62,76,52,66,32,37)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 460);