Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#7878 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C17 | C2×D4 |
| commutator | C2×C4 | C2×C42×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| socle | C23×C17 | C2×C4⋊C4 |
| prime radical | C24×C17 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C17 ◁ C22×C42×C17 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23)(24,32)(25,29)(26,30)(27,31)(28,33)(34,37)(35,36)(38,39,41,44,42,45,48,51)(40,43,47,50,46,49,52,53);(19,21)(22,24,26,34,27,35,25,33)(23,28,29,36,31,37,30,32)(38,40)(39,43)(41,47)(42,46)(44,50)(45,49)(48,52)(51,53);(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32)(40,46)(43,49)(47,52)(50,53)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 866);