Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C73×#60472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | C22×#121 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C2×#41×C73 ◁ G | |
2578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 137 points:
(1,2,8,14)(3,15,20,31)(4,21,24,38)(5,11,25,35)(6,13,7,12)(9,19,32,18)(10,28,34,23)(16,41,44,29)(17,40,45,55)(22,49,50,58)(26,37,27,36)(30,47,54,46)(33,52,57,51)(39,48,56,43)(42,60,61,53)(59,63,64,62)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3,7,19,8,20,6,18)(2,9,13,31,14,32,12,15)(4,16,28,48,24,44,23,43)(5,17,27,47,25,45,26,46)(10,29,38,56,34,41,21,39)(11,30,37,55,35,54,36,40)(22,42,52,63,50,61,51,62)(33,53,58,64,57,60,49,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,28)(8,24)(9,29)(11,33)(12,21)(13,38)(14,34)(15,39)(17,42)(18,43)(19,48)(20,44)(25,50)(26,51)(27,52)(30,53)(31,56)(32,41)(35,57)(36,49)(37,58)(40,59)(45,61)(46,62)(47,63)(54,60)(55,64);(1,5)(2,11)(3,17)(4,22)(6,26)(7,27)(8,25)(9,30)(10,33)(12,36)(13,37)(14,35)(15,40)(16,42)(18,46)(19,47)(20,45)(21,49)(23,51)(24,50)(28,52)(29,53)(31,55)(32,54)(34,57)(38,58)(39,59)(41,60)(43,62)(44,61)(48,63)(56,64);(5,25)(11,35)(17,45)(22,50)(26,27)(30,54)(33,57)(36,37)(40,55)(42,61)(46,47)(49,58)(51,52)(53,60)(59,64)(62,63)4234 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 53460);