Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#7470 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C17 | D4 |
| commutator | C4 | C4×C16×C17 |
| Frattini, cosocle | C22×C16 | C22×C17 |
| socle | C22×C17 | C22⋊C16 |
| prime radical | C23×C17 | M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C17 ◁ C22×C16×C17 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(22,23,26,32,27,33,45,58,28,34,46,59,48,61,73,81,29,35,47,60,49,62,74,82,25,31,42,55,43,56,70,79)(24,36,38,63,39,64,67,83,37,53,65,57,66,77,84,80,40,54,44,71,68,78,72,85,41,30,50,51,69,52,75,76);(18,19,20,21)(22,24)(23,30)(25,41)(26,44)(27,39)(28,37)(29,40)(31,54)(32,57)(33,52)(34,36)(35,53)(38,47)(42,65)(43,69)(45,72)(46,50)(48,66)(49,68)(51,60)(55,63)(56,78)(58,80)(59,71)(61,64)(62,77)(67,74)(70,84)(73,75)(76,82)(79,83)(81,85)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 458);