Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C28 |
| direct product | C73×#62611 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| prime radical | C73×#3552 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ #529×C73 ◁ G | |
1646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 9344 |
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)(17,18,22,26)(19,23,28,36)(20,30,31,45)(21,33,34,25)(24,38,39,32)(27,40,41,48)(29,44,43,37)(35,46,47,42)(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)(17,19,22,28)(18,23,26,36)(20,27,31,41)(21,29,34,43)(24,35,39,47)(25,37,33,44)(30,40,45,48)(32,42,38,46);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,32,22,31,34,38)(18,24,25,30,26,39,33,45)(19,27,29,42,28,41,43,46)(23,35,37,40,36,47,44,48);(3,12)(7,10)(11,13)(15,16)(20,31)(24,39)(27,41)(30,45)(32,38)(35,47)(40,48)(42,46);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,28)(20,31)(23,36)(24,39)(29,43)(30,45)(32,38)(37,44)3577 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 55599);