Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C73 · Schur multiplier (H2) = C25×C4 |
| direct product | C4×#442×C73 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | C2×D4 |
| commutator | C4 | C22×C42×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C23×C73 | C22×D4 |
| prime radical | C22×D8×C73 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ C43×C73 ◁ G | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,6,8,7)(9,10,12,14)(11,13,16,15);(10,13)(11,16)(14,15);(5,7,8,6)(10,14)(13,15)6424 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 13488);