Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8×C17 · Schur multiplier (H2) = C23×C42 |
| direct product | C4×C17×#336 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42×C17 | C22 |
| commutator | C2 | C42×C8×C17 |
| Frattini, cosocle | C23×C4 | C23×C17 |
| socle, prime radical | C24×C17 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42×C17 ◁ G | |
1318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 4352 |
A minimal faithful permutation representation — 3 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)(7,8)(13,14,15,17,16,18,19,20);(6,8)(9,10,11,12)2720 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 3680);