Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C17 · Schur multiplier (H2) = C27 |
| direct product | C22×C17×#331 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C17 | C22⋊C4 |
| commutator | C2×C4 | C23×C4×C17 |
| Frattini, cosocle | C42 | C24×C17 |
| prime radical | C22×C17×#127 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C17 ◁ C22×C42×C17 ◁ G | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4);(5,6,7,10,8,9,11,12)(13,14)(15,17)(16,19)(18,20);(6,9)(7,11)(14,17)(15,20)(18,19)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 10681);