Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#13157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1936 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C23×C17 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,24,31,26,32,25,54)(20,33,37,56,38,65,52,66)(21,39,43,61,44,67,53,75)(22,45,23,49,48,30,47,29)(27,36,57,35,59,50,58,46)(28,41,62,42,64,76,63,51)(34,69,72,80,73,55,40,60)(68,77,78,71,81,70,79,74);(18,20,22,46,25,37,47,36,26,38,48,35,24,52,23,50)(19,27,49,33,31,58,30,66,32,59,29,65,54,57,45,56)(21,40,76,71,43,73,51,77,44,72,41,74,53,34,42,70)(28,60,61,79,63,69,39,68,64,80,75,78,62,55,67,81);(18,21,22,41,25,43,47,42,26,44,48,76,24,53,23,51)(19,28,29,61,31,63,45,39,32,64,49,75,54,62,30,67)(20,34,35,70,37,40,50,71,38,73,46,77,52,72,36,74)(27,55,33,68,58,60,66,78,59,69,65,81,57,80,56,79)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6145);