Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#12608 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | #117 |
| commutator | C2×C4 | C2×C42×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | #432 |
| prime radical | C24×C17 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C17×#97 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
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)(24,34,37,45)(25,31)(26,32)(27,33)(28,30)(29,36,42,35)(38,48,51,53)(39,46)(40,44)(41,47)(43,50,52,49);(19,21)(22,24,26,38,27,36,41,50,28,37,40,51,25,35,39,49)(23,29,32,43,33,34,47,48,30,42,44,52,31,45,46,53);(23,30)(24,35,37,36)(26,40)(29,34,42,45)(31,33)(38,50,51,49)(39,41)(43,53,52,48)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5596);