Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#7469 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C17 | C22⋊C4 |
| commutator | C22 | C4×C16×C17 |
| Frattini, cosocle | C22×C16 | C22×C17 |
| socle | C2×C17 | #1374 |
| prime radical | C23×C17 | C2×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C17 ◁ C22×C16×C17 ◁ G | |
174 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)(18,19,21,26,22,28,35,46,23,29,36,47,38,50,59,70,24,27,37,45,39,48,60,68,40,49,61,69,62,71,77,80)(20,30,32,51,33,52,56,72,34,53,57,73,58,74,76,81,31,25,41,42,54,43,63,65,55,44,64,66,75,67,78,79);(18,20)(19,25,27,30)(21,32)(22,33)(23,34)(24,31)(26,42,45,51)(28,43,48,52)(29,44,49,53)(35,56)(36,57)(37,41)(38,58)(39,54)(40,55)(46,65,68,72)(47,66,69,73)(50,67,71,74)(59,76)(60,63)(61,64)(62,75)(70,79,80,81)(77,78)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 457);