Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C17 · Schur multiplier (H2) = C26×C42 |
| direct product | C4×C17×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C17 | C24 |
| commutator | C22 | C24×C4×C17 |
| Frattini, cosocle | C23 | C25×C17 |
| socle | C23×C17 | C25 |
| prime radical | C2×C17×H(3,𝔽4) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C17 ◁ G | |
2664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(5,6,9,12)(7,14,15,11)(8,16,18,20)(10,13,19,17);(5,7,9,15)(6,11,12,14)(8,17,18,13)(10,16,19,20);(5,8)(6,7)(9,18)(10,11)(12,15)(13,16)(14,19)(17,20);(5,7)(6,8)(9,15)(10,16)(11,13)(12,18)(14,17)(19,20)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 27341);