Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C73 · Schur multiplier (H2) = C210 |
| direct product | C23×#103×C73 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C73 | D4 |
| commutator | C4 | C24×C4×C73 |
| Frattini, cosocle | C2×C4 | C25×C73 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C73 ◁ C24×C4×C73 ◁ G | |
5502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 5 generators on 91 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91);(3,4);(5,6);(7,8,9,10)(12,15)(13,17)(16,18);(11,12,14,16)(13,18,17,15)8176 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53066);