Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#12942 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #447 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1862 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×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)(18,19,24,31,26,32,25,47)(20,33,37,51,38,64,48,60)(21,39,42,56,43,76,49,75)(22,30,45,61,46,62,23,29)(27,44,52,72,54,36,53,35)(28,55,57,77,59,40,58,41)(34,65,68,50,70,79,69,80)(63,73,81,71,78,66,74,67);(18,20,24,37,26,38,25,48)(19,27,31,52,32,54,47,53)(21,34,42,68,43,70,49,69)(22,44,45,72,46,36,23,35)(28,50,57,79,59,80,58,65)(29,60,30,33,61,51,62,64)(39,74,56,63,76,81,75,78)(40,71,41,66,55,67,77,73);(18,21)(19,28)(20,34)(22,41)(23,40)(24,42)(25,49)(26,43)(27,50)(29,39)(30,56)(31,57)(32,59)(33,63)(35,71)(36,73)(37,68)(38,70)(44,66)(45,55)(46,77)(47,58)(48,69)(51,81)(52,79)(53,65)(54,80)(60,74)(61,76)(62,75)(64,78)(67,72)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5930);