Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C17 · Schur multiplier (H2) = C27 |
| direct product | C22×C17×D4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C17 | D4 |
| commutator | C4 | C23×C8×C17 |
| Frattini, cosocle | C42 | C24×C17 |
| socle | C24×C17 | C22⋊C4 |
| prime radical | C23×D4×C17 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C17 ◁ C22×C42×C17 ◁ G | |
1886 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)(7,9)(8,11)(10,12)(13,14,15,17,16,18,19,20);(6,9)(7,12)(10,11)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 10419);