Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#33542 |
| 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 | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,8,14)(3,15,19,43)(4,20,24,35)(5,13,7,11)(6,26,29,38)(9,31,34,18)(10,21,37,23)(12,25,40,28)(16,44,48,62)(17,49,30,39)(22,53,56,36)(27,41,57,33)(32,54,60,64)(42,45,63,47)(46,50,58,59)(51,52,61,55)(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)(2,9)(4,16)(5,17)(6,27)(7,30)(8,19)(10,32)(11,33)(12,39)(13,41)(14,34)(15,26)(18,28)(20,50)(21,45)(22,54)(23,47)(24,48)(25,31)(29,57)(35,59)(36,62)(37,60)(38,43)(40,49)(42,51)(44,53)(46,55)(52,58)(56,64)(61,63);(1,4)(2,10)(3,16)(5,21)(6,22)(7,23)(8,24)(9,32)(11,35)(12,36)(13,20)(14,37)(15,42)(17,45)(18,46)(19,48)(25,52)(26,51)(27,54)(28,55)(29,56)(30,47)(31,58)(33,59)(34,60)(38,61)(39,62)(40,53)(41,50)(43,63)(44,49)(57,64);(1,5,8,7)(2,11,14,13)(3,17,19,30)(4,21,24,23)(6,25,29,28)(9,33,34,41)(10,35,37,20)(12,38,40,26)(15,39,43,49)(16,45,48,47)(18,27,31,57)(22,52,56,55)(32,59,60,50)(36,61,53,51)(42,62,63,44)(46,54,58,64)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26530);