Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4×C17 · Schur multiplier (H2) = C214 |
| direct product | C17×#62652 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | C26 |
| commutator | C2 | C25×C4×C17 |
| Frattini, cosocle | C22 | C26×C17 |
| prime radical | C22×C17×H(5,𝔽2) | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ G | |
11674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 6 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,21)(22,23)(24,28)(25,32)(26,27)(29,33)(30,34)(31,37)(35,36);(23,27)(28,34)(29,32)(35,37);(24,30)(28,34)(31,36)(35,37);(22,24)(23,28)(25,31)(26,30)(27,34)(29,35)(32,37)(33,36);(22,23)(24,28)(25,29)(26,27)(30,34)(31,35)(32,33)(36,37);(22,25)(23,29)(24,31)(26,33)(27,32)(28,35)(30,36)(34,37)2210 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 55640);