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×#12830 |
| 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)(18,19,23,30,25,28,24,47)(20,31,34,52,35,26,46,51)(21,36,39,68,37,69,48,58)(22,40,43,56,45,72,44,29)(27,42,53,70,55,76,54,38)(32,60,63,81,61,77,64,79)(33,65,67,57,41,73,75,50)(49,66,78,62,59,74,80,71);(18,20,24,34,25,35,23,46)(19,26,47,51,28,31,30,52)(21,32,48,63,37,61,39,64)(22,41,44,75,45,33,43,67)(27,49,54,78,55,59,53,80)(29,57,72,73,56,50,40,65)(36,60,58,81,69,77,68,79)(38,71,76,66,70,62,42,74);(18,21)(19,27)(20,32)(22,42)(23,39)(24,48)(25,37)(26,49)(28,55)(29,58)(30,53)(31,59)(33,66)(34,63)(35,61)(36,40)(38,44)(41,74)(43,70)(45,76)(46,64)(47,54)(50,79)(51,78)(52,80)(56,68)(57,81)(60,65)(62,67)(69,72)(71,75)(73,77)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5818);