Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#12642 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D8 |
| commutator | C8 | C2×C42×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C4×D8 |
| prime radical | C2×C17×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
430 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)(22,23,26,33,29,34,28,44)(24,35,38,48,41,53,40,46)(25,31,42,50,43,51,27,32)(30,45,47,36,49,39,52,37);(19,21)(22,24,27,39,28,40,43,36,29,41,42,45,26,38,25,37)(23,30,31,48,33,47,50,53,34,49,51,46,44,52,32,35);(22,25,26,42,29,43,28,27)(23,31,33,50,34,51,44,32)(24,36,38,39,41,37,40,45)(30,46,47,35,49,48,52,53)1088 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5630);