Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C17 · Schur multiplier (H2) = C24 |
| direct product | C17×#12549 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C2×D4 |
| commutator | C4 | C2×C4×C8×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C2×C4⋊C8 |
| prime radical | C24×C17 | M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ C23×C8×C17 ◁ G | |
586 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)(22,23,26,31,27,30,40,45,28,32,41,46,25,38,39,51)(24,33,36,48,37,43,50,52,34,47,49,53,35,29,42,44);(19,21)(22,24)(23,29,32,43)(25,35)(26,36)(27,37)(28,34)(30,33,38,47)(31,44,46,52)(39,42)(40,50)(41,49)(45,48,51,53);(24,34)(29,43)(33,47)(35,37)(36,49)(42,50)(44,52)(48,53)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5537);