Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12952 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#12954 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1864 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| prime radical | C23×C17 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
380 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)(18,19,23,28,24,30,48,29)(20,31,35,63,36,51,46,58)(21,37,39,59,40,75,47,74)(22,41,44,27,45,60,69,61)(25,49,52,33,54,42,53,34)(26,43,55,38,57,76,56,77)(32,64,67,81,68,50,72,80)(62,71,79,66,73,70,78,65);(18,20,24,36)(19,25)(21,32,40,68)(22,42,45,49)(23,46,48,35)(26,50)(27,58)(28,53)(29,52)(30,54)(31,41)(33,69,34,44)(37,73)(38,71,77,70)(39,72,47,67)(43,66,76,65)(51,60)(55,81)(56,80)(57,64)(59,79)(61,63)(62,75)(74,78);(18,21,24,40)(19,26,30,57)(20,32,36,68)(22,43,45,76)(23,47,48,39)(25,50,54,64)(27,59,61,74)(28,56,29,55)(31,62,51,73)(33,70,34,71)(35,72,46,67)(37,41,75,60)(38,44,77,69)(42,66,49,65)(52,81,53,80)(58,79,63,78)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5942);