Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×#7462 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | D16 |
| commutator | C16 | C2×C8×C17 |
| Frattini, cosocle | C4×C16 | C22×C17 |
| socle | C2×C17 | #1371 |
| prime radical | C22×C17 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C2×C8×C17 ◁ C4×C8×C17 ◁ C4×C16×C17 ◁ G | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,28,24,29,43,55,25,31,44,56,46,30,23,45)(20,32,35,58,37,59,67,79,38,61,68,80,70,60,36,69)(21,39,40,52,41,71,64,54,42,73,74,53,63,72,33,27)(26,47,48,65,49,75,77,66,51,76,78,81,50,62,57,34);(18,20,33,62,23,36,63,81,46,70,74,76,44,68,42,66,25,38,64,75,43,67,41,65,24,37,40,47,22,35,21,34)(19,26,39,58,28,48,52,59,29,49,71,79,55,77,54,61,31,51,73,80,56,78,53,60,30,50,72,69,45,57,27,32)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 450);