Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C1 |
| direct product | C17×#7454 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1365 |
| commutator | C2×C16 | C2×C4×C17 |
| Frattini, cosocle | C4×C16 | C22×C17 |
| prime radical | C22×C17 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C4×C8×C17 ◁ C4×C16×C17 ◁ G | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 2 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,29,47,31,42,28,23,50,48,60,53,59,45,79,26,32,43,64,24,51,44,61,25,30,49,63,41,66,21,40)(20,33,35,55,74,58,73,67,36,62,75,27,52,54,34,56,39,70,77,65,37,78,72,68,38,69,76,57,71,81,46,80);(18,20)(19,27,32,57)(21,35)(22,46)(23,38)(24,52)(25,36)(26,39)(28,58,61,78)(29,62,64,69)(30,65,50,55)(31,67,51,68)(33,63,70,60)(34,43)(37,53)(40,54,79,81)(41,74)(42,76)(44,75)(45,77)(47,71)(48,72)(49,73)(56,59,80,66)629 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 442);