Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1790 |
| commutator | C2×Q8 | C22×C4×C17 |
| Frattini, cosocle | C22×Q8 | C23×C17 |
| prime radical | C17×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C23×C17 ◁ C22×Q8×C17 ◁ G | |
730 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)(18,19,24,30,25,31,23,29)(20,32,34,48,35,46,40,42)(21,36,38,27,22,28,37,45)(26,41,43,33,44,49,47,39);(18,20)(19,26)(21,33)(22,39)(23,40)(24,34)(25,35)(27,42)(28,46)(29,43)(30,47)(31,44)(32,36)(37,41)(38,49)(45,48);(18,21,25,22)(19,27,31,45)(20,33,35,39)(23,38,24,37)(26,42,44,48)(28,30,36,29)(32,43,46,47)(34,41,40,49)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5770);