Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#12829 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #418 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1804 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
378 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)(20,30,33,25)(21,35)(22,39)(23,45)(24,27)(26,37)(28,43)(29,42)(31,58,59,77)(32,62,40,73)(34,49,65,57)(36,68)(38,71)(41,55)(44,72)(46,54)(47,60,56,70)(48,66,53,75)(50,76)(51,69)(52,63)(61,79,67,81)(64,78,74,80);(18,20,23,34,24,33,42,65)(19,25,45,57,27,30,29,49)(21,31,46,67,36,59,38,61)(22,40,43,75,44,32,63,66)(26,47,50,78,51,56,55,80)(28,53,72,73,52,48,39,62)(35,58,54,81,68,77,71,79)(37,70,76,64,69,60,41,74);(18,21)(19,26)(20,31)(22,41)(23,46)(24,36)(25,47)(27,51)(28,54)(29,55)(30,56)(32,64)(33,59)(34,67)(35,39)(37,43)(38,42)(40,74)(44,76)(45,50)(48,79)(49,80)(52,71)(53,81)(57,78)(58,62)(60,66)(61,65)(63,69)(68,72)(70,75)(73,77)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5817);