Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#33532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C73 |
| prime radical | C73×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×C4×C73 ◁ C2×#131×C73 ◁ G | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,7,12)(3,13,16,42)(4,9,18,33)(5,21,26,56)(6,27,24,11)(8,29,31,15)(10,36,39,25)(14,43,44,41)(17,48,28,37)(19,50,23,55)(20,53,49,35)(22,40,58,32)(30,47,60,45)(34,59,38,52)(46,64,54,62)(51,63,57,61)(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)(2,8)(4,14)(5,22)(6,28)(7,16)(9,30)(10,37)(11,40)(12,31)(13,21)(15,25)(17,24)(18,44)(19,51)(20,54)(23,57)(26,58)(27,32)(29,36)(33,60)(34,62)(35,63)(38,64)(39,48)(41,55)(42,56)(43,50)(45,52)(46,49)(47,59)(53,61);(1,4)(2,9)(3,14)(5,23)(6,20)(7,18)(8,30)(10,38)(11,35)(12,33)(13,41)(15,47)(16,44)(17,46)(19,26)(21,55)(22,57)(24,49)(25,59)(27,53)(28,54)(29,45)(31,60)(32,61)(34,39)(36,52)(37,64)(40,63)(42,43)(48,62)(50,56)(51,58);(4,18)(9,33)(14,44)(19,23)(20,49)(30,60)(34,38)(35,53)(41,43)(45,47)(46,54)(50,55)(51,57)(52,59)(61,63)(62,64)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26520);