Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#9048 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | #120 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| socle | C22×C17 | #386 |
| prime radical | C23×C17 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C42×C17 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 2176 |
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)(18,19,22,24,23,27,21,26)(20,28,30,25,31,32,29,33)(34,35,38,42,39,36,40,44)(37,43,46,49,45,41,47,48);(19,24)(21,22)(25,28)(26,27)(29,30)(32,33)(34,36,38,44,39,35,40,42)(37,43,47,48,45,41,46,49);(18,20,23,31)(19,25,27,33)(21,30,22,29)(24,28,26,32)(34,37)(35,41)(36,43)(38,47)(39,45)(40,46)(42,49)(44,48)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 2036);