Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#12890 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C17 | C2×D8 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C17 ◁ C2×C42×C17 ◁ C2×C4×C8×C17 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 4352 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21)(22,23)(24,27)(25,26)(28,29)(30,31,36,32)(33,35,41,38)(34,42,45,39)(37,43,44,40);(23,26)(27,29)(30,32,35,42,37,43,34,41,36,31,38,39,44,40,45,33);(22,24,25,28)(23,27,26,29)(30,33,34,43,44,39,35,32,36,41,45,40,37,42,38,31)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5878);