Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#13016 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #447 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1879 |
| prime radical | C2×C17×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
462 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)(18,19,24,30,25,31,47,63,26,32,23,48,46,62,22,29)(20,33,37,56,38,61,35,64,39,67,36,27,51,54,49,55)(21,40,42,65,43,76,41,28,44,58,45,59,52,57,50,60)(34,68,70,75,71,53,73,66,72,79,77,81,74,80,69,78);(18,20,25,51,26,39,46,38)(19,27,31,64,32,56,62,55)(21,34,52,71,44,72,43,74)(22,36,24,35,47,37,23,49)(28,53,65,79,60,80,59,68)(29,61,30,33,63,54,48,67)(40,75,57,66,58,81,76,78)(41,70,42,73,50,77,45,69);(18,21)(19,28)(20,34)(22,45)(23,50)(24,41)(25,52)(26,44)(27,53)(29,40)(30,57)(31,65)(32,60)(33,66)(35,70)(36,69)(37,73)(38,74)(39,72)(42,47)(43,46)(48,76)(49,77)(51,71)(54,81)(55,68)(56,80)(58,63)(59,62)(61,75)(64,79)(67,78)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6004);