Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C73×#61053 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ #525×C73 ◁ G | |
2666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2)(3,4)(5,6)(7,15)(8,12)(9,22)(10,13)(11,17)(14,25)(16,33)(18,31)(19,36)(20,32)(21,29)(23,27)(24,30)(26,35)(28,34)(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);(5,7,10,18)(6,11,13,27)(8,19,21,35)(9,23,25,17)(12,28,29,33)(14,31,22,15)(16,24,34,20)(26,32,36,30);(2,4);(5,8,10,21)(6,12,13,29)(7,16,18,34)(9,24,25,20)(11,26,27,36)(14,32,22,30)(15,33,31,28)(17,35,23,19);(6,13)(11,27)(12,29)(14,22)(15,31)(26,36)(28,33)(30,32)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 54041);