Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C17×C64 · Schur multiplier (H2) = C22 |
| direct product | C17×C22⋊C64 |
| unambiguous semidirect product | C22⋊(C17×C64) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C17×C32 | C22 |
| commutator | C2 | C2×C17×C64 |
| Frattini, cosocle | C2×C32 | C22×C17 |
| socle | C22×C17 | C2×C32 |
| prime radical | C23×C17 | C32 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17×C32 ◁ G | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,24,29,25,30,34,44,26,31,35,45,38,48,54,64,27,32,36,46,39,49,55,65,41,51,57,67,60,70,74,79,28,33,37,47,40,50,56,66,42,52,58,68,61,71,75,80,43,53,59,69,62,72,76,81,63,73,77,82,78,83,84,85);(19,21)2720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 500);