Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#21545 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C73 | C22×D4 |
| commutator | C2×C4 | C23×C4×C73 |
| Frattini, cosocle | C22×C4 | C24×C73 |
| socle | C22×C73 | C2×C4×D4 |
| prime radical | C73×#3050 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C73 ◁ C22×M16×C73 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 109 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,19,26,20,30,18,12)(8,13,23,28,21,27,22,34)(17,29,33,35,31,36,32,25)(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);(5,7,9,18,11,20,10,19)(6,12,14,16,15,26,24,30)(8,17,22,32,21,31,23,33)(13,25,28,29,27,35,34,36);(2,4)(8,21)(13,27)(17,31)(22,23)(25,35)(28,34)(29,36)(32,33);(5,8,10,23,11,21,9,22)(6,13,14,28,15,27,24,34)(7,17,19,33,20,31,18,32)(12,25,16,29,26,35,30,36)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 14533);