Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#7475 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #367 |
| commutator | C16 | C42×C17 |
| Frattini, cosocle | C22×C16 | C22×C17 |
| socle | C2×C17 | #1380 |
| prime radical | C23×C17 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C22×C8×C17 ◁ C22×C16×C17 ◁ G | |
170 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)(20,32,34,60)(21,28)(22,42)(23,45)(24,48)(25,31)(26,33,49,63)(27,39)(29,47)(30,46)(35,68,64,59)(36,70,66,81)(37,72,67,61)(38,55)(40,75)(41,58)(43,53)(44,54)(50,62,77,71)(51,79,78,69)(52,80,76,65)(56,74)(57,73);(18,20,22,35,43,66,73,80,46,67,74,79,47,71,41,63,25,34,39,64,23,36,40,65,24,37,44,69,38,62,21,33)(19,26,28,50,55,78,54,61,48,76,75,81,45,68,27,32,31,49,58,77,29,51,56,72,30,52,57,70,53,59,42,60)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 463);