Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#8056 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C17 |
| Frattini, cosocle | C23×C4 | C23×C17 |
| socle | C22×C17 | C2×#95 |
| prime radical | C24×C17 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C42×C17 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1088 |
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)(18,19,21,26,23,27,22,25)(20,28,30,24,29,33,31,32)(34,35)(36,42)(37,40)(38,41)(39,44)(43,48)(45,47)(46,49);(18,20,22,31,23,29,21,30)(19,24,26,33,27,32,25,28)(34,36)(35,39,41,42)(37,43)(38,44)(40,46,47,48)(45,49);(20,29)(24,32)(28,33)(30,31)(34,37,38,45)(35,40,41,47)(36,43,44,49)(39,46,42,48)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 1044);