Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C17×C32 · Schur multiplier (H2) = C1 |
| direct product | C17×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C17×C32 | D4 |
| commutator | C4 | C2×C17×C32 |
| Frattini, cosocle | C2×C32 | C22×C17 |
| socle | C2×C17 | C4⋊C32 |
| prime radical | C22×C17 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C17×C32 ◁ C2×C17×C32 ◁ G | |
70 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)(18,19,22,29,23,30,41,54,24,31,42,55,45,57,68,76,25,28,43,51,46,52,69,73,37,53,63,74,64,75,79,81,26,32,44,56,47,58,70,77,34,27,48,49,61,50,71,72,21,38,39,65,40,66,67,80,20,33,35,59,36,60,62,78);(18,20,21,34,26,37,25,24)(19,27,28,33,32,31,38,53)(22,35,39,48,44,63,43,42)(23,36,40,61,47,64,46,45)(29,49,51,59,56,55,65,74)(30,50,52,60,58,57,66,75)(41,62,67,71,70,79,69,68)(54,72,73,78,77,76,80,81)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 532);