Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C73×#60462 |
| 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 ◁ C22×#41×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)(5,6,10,14)(7,15,17,28)(8,12,20,26)(9,22,16,13)(11,23,24,19)(18,29,31,36)(21,33,30,27)(25,34,35,32)(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);(5,7,16,23,10,17,9,19)(6,11,22,28,14,24,13,15)(8,18,30,34,20,31,21,32)(12,25,33,36,26,35,27,29);(2,4)(7,17)(11,24)(15,28)(18,31)(19,23)(25,35)(29,36)(32,34);(5,8)(6,12)(7,18)(9,21)(10,20)(11,25)(13,27)(14,26)(15,29)(16,30)(17,31)(19,32)(22,33)(23,34)(24,35)(28,36);(8,20)(12,26)(18,31)(21,30)(25,35)(27,33)(29,36)(32,34)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53450);