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×#12715 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1781 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C2×#41 | C23×C17 |
| prime radical | C17×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C2×C17×2−1+4 ◁ G | |
976 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)(20,29,32,42)(21,26,33,24)(22,27)(23,28)(25,31,40,35)(30,45)(34,41,48,38)(36,43)(37,47)(39,49)(44,46);(19,24)(21,33)(25,37)(26,28)(27,38)(29,44)(30,46)(34,48)(39,40)(41,43)(42,45)(47,49);(18,20)(19,25)(21,30)(22,35)(23,32)(24,37)(26,39)(27,42)(28,40)(29,43)(31,36)(33,46)(34,49)(38,45)(41,44)(47,48)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5703);