Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#13558 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C17 | C2×D4 |
| commutator | C4 | C22×C16×C17 |
| Frattini, cosocle | C2×C16 | C23×C17 |
| socle | C2×C17 | C2×C22⋊C16 |
| prime radical | C2×#41×C17 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C17 ◁ C22×C16×C17 ◁ G | |
290 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)(18,19,22,29,23,30,41,56,24,31,42,57,44,58,69,78,25,27,43,50,45,51,70,74,21,28,38,53,39,54,66,76)(20,32,34,59,35,60,63,79,33,47,61,55,62,72,80,77,36,52,40,67,64,75,68,81,37,26,46,48,65,49,71,73);(18,20)(19,26)(21,37)(22,40)(23,35)(24,33)(25,36)(27,47)(28,52)(29,55)(30,49)(31,32)(34,43)(38,61)(39,65)(41,68)(42,46)(44,62)(45,64)(48,50)(51,72)(53,59)(54,75)(56,77)(57,67)(58,60)(63,70)(66,80)(69,71)(73,74)(76,79)(78,81);(19,27)(26,47)(28,31)(29,50)(30,51)(32,52)(48,55)(49,72)(53,57)(54,58)(56,74)(59,67)(60,75)(73,77)(76,78)(79,81)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6546);