Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#13195 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C2×D4 |
| commutator | C2×C4 | C22×C8×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C8×D4 |
| prime radical | C2×C17×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ C22×C16×C17 ◁ G | |
438 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)(19,21)(22,23,28,34,29,33,43,49,26,32,36,31,27,44,25,42)(24,35,40,46,41,50,52,53,38,30,45,48,39,47,37,51);(18,19,20,21)(22,24,36,45,29,41,25,37,26,38,28,40,27,39,43,52)(23,30,31,46,33,47,42,53,32,35,34,48,44,50,49,51);(22,25,27,36,26,43,29,28)(23,31,33,42,32,34,44,49)(24,37,39,45,38,52,41,40)(30,46,47,53,35,48,50,51)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6183);