Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#13230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1964 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×#41 | C23×C17 |
| prime radical | C2×#41×C17 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×C17×#131 ◁ G | |
436 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)(20,30,35,53)(21,36)(22,41,31,27)(23,44)(24,29)(25,33)(26,39)(28,37)(32,56,60,79)(34,62,57,52)(38,67,54,64)(40,65)(42,47)(43,50,55,71)(45,74)(46,72)(48,69,70,78)(49,66)(51,76,77,59)(58,63)(61,73)(68,80)(75,81);(18,20,31,33,24,35,22,42)(19,25,41,53,29,47,27,30)(21,32,54,58,40,60,38,68)(23,34,55,61,37,57,43,72)(26,45,70,77,49,75,48,51)(28,46,71,62,44,73,50,52)(36,63,67,79,65,80,64,56)(39,59,78,74,66,76,69,81);(18,21,37,66,24,40,23,39)(19,26,44,65,29,49,28,36)(20,32,57,76,35,60,34,59)(22,38,55,78,31,54,43,69)(25,45,73,80,47,75,46,63)(27,48,71,67,41,70,50,64)(30,51,62,79,53,77,52,56)(33,58,72,81,42,68,61,74)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6218);