Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#12678 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #442 |
| commutator | C8 | C2×C42×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1776 |
| prime radical | C23×C17 | #137 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#97 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,29,24,30,38,28,25,31,39,45,23,27,21,37)(20,32,34,43,35,44,48,47,36,46,49,26,40,42,33,41);(18,20,23,35,25,36,24,40)(19,26,27,41,31,43,30,47)(21,34,39,48,38,49,22,33)(28,44,29,46,37,42,45,32);(19,27,31,30)(21,38)(22,39)(26,41,43,47)(28,45,37,29)(32,42,46,44)(33,48)(34,49)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5666);