Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C73 · Schur multiplier (H2) = C28 |
| direct product | C73×#34348 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C73 | C24 |
| commutator | C22 | C24×C4×C73 |
| Frattini, cosocle | C23 | C25×C73 |
| socle | C23×C73 | C25 |
| prime radical | C24×C73 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C73 ◁ G | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2,3,4)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,31)(25,28)(27,33)(29,36)(30,35)(32,34)(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)(6,10)(8,15)(9,14)(11,13)(12,17)(16,20)(18,19)(21,23,25,30)(22,26,28,35)(24,32,33,29)(27,36,31,34);(7,14)(10,13)(15,20)(17,19)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36);(6,11)(7,14)(12,18)(15,20)(22,28)(26,35)(27,31)(34,36);(5,8,9,16)(6,12,11,18)(7,15,14,20)(10,17,13,19)5548 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 27336);