Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1781 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×#41 | C23×C17 |
| prime radical | C17×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×C17×2−1+4 ◁ G | |
944 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)(20,30)(21,33,35,44)(22,27,37,40)(23,28)(24,29)(25,32,41,36)(26,31,43,38)(34,46)(39,48)(42,47)(45,49);(18,20,24,23)(19,25,29,41)(21,31,35,38)(22,36,37,32)(26,39,43,46)(27,28,40,30)(33,42,44,49)(34,47,48,45);(18,21)(19,26)(20,31)(22,34)(23,38)(24,35)(25,39)(27,42)(28,44)(29,43)(30,33)(32,45)(36,47)(37,48)(40,49)(41,46)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5705);