Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C28 |
| direct product | C73×#62600 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C73×#3562 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ #529×C73 ◁ G | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 121 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,41)(29,38)(30,39)(31,45)(33,40)(34,46)(37,43)(42,47)(44,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,41)(33,42)(34,44)(35,43)(36,45)(40,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,35,23,34,21,33)(18,25,26,32,27,40,36,46)(19,28,31,43,30,44,29,42)(24,37,38,41,39,47,45,48);(3,12)(7,10)(11,13)(15,16)(17,21,23,22)(18,26,27,36)(19,29,30,31)(20,33,34,35)(24,38,39,45)(25,32,40,46)(28,42,44,43)(37,41,47,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,30)(20,34)(24,39)(25,40)(29,31)(32,46)(33,35)(38,45)3577 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 55588);