Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#13126 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C17 | #124 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1924 |
| prime radical | C2×#41×C17 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C17 ◁ C22×C8×C17 ◁ G | |
300 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)(18,19,23,29,24,30,47,62,25,31,48,63,22,28,33,50)(20,32,37,64,35,54,65,60,38,52,45,76,36,26,49,53)(21,39,43,72,42,59,66,61,44,57,46,77,41,27,55,58)(34,67,68,79,40,73,74,81,69,51,71,78,70,56,75,80);(18,20,33,65,22,36,48,37,25,38,47,49,24,35,23,45)(19,26,50,64,28,52,63,53,31,54,62,76,30,32,29,60)(21,40,55,71,41,34,46,74,44,70,66,68,42,69,43,75)(27,56,77,79,57,51,61,80,59,73,72,78,39,67,58,81);(18,21,22,41,25,44,24,42)(19,27,28,57,31,59,30,39)(20,34,35,40,38,69,36,70)(23,46,33,66,48,43,47,55)(26,51,32,56,54,67,52,73)(29,61,50,72,63,58,62,77)(37,71,65,75,45,68,49,74)(53,79,64,81,60,78,76,80)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6114);