Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C22×C4 |
| direct product | C73×#8817 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C73 | #127 |
| commutator | C42 | C22×C4×C73 |
| Frattini, cosocle | C2×C42 | C23×C73 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C73 ◁ C2×C42×C73 ◁ G | |
390 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 18688 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,3,4)(5,6)(7,13)(8,17)(9,11)(10,15)(12,14)(16,19)(18,20)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(5,7,14,20,9,16,8,15)(6,10,17,19,11,18,12,13)(22,26)(27,34)(28,33)(29,35);(6,11)(10,18)(12,17)(13,19)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 1805);