Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C17 · Schur multiplier (H2) = C29 |
| direct product | C22×C17×#443 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C17 | D4 |
| commutator | C4 | C24×C4×C17 |
| Frattini, cosocle | C2×C4 | C25×C17 |
| socle | C23×C17 | C4×D4 |
| prime radical | C22×C17×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C17 ◁ C23×C8×C17 ◁ G | |
2750 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)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4);(5,6,9,13,10,7,14,12)(8,11,16,19,17,15,20,18);(5,7)(6,10)(8,15)(9,12)(11,17)(13,14)(16,18)(19,20);(5,8,9,16,10,17,14,20)(6,11,13,19,7,15,12,18)1904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 53165);