Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C26 |
| direct product | C73×#24423 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C73×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×C4×C73 ◁ G | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,8,30,7,12,6,13)(3,14,28,33,18,9,17,39)(4,19,29,10,22,40,23,36)(5,24,26,37,27,11,25,38)(15,43,49,41,45,34,20,31)(16,32,47,58,48,42,46,57)(21,50,54,55,52,60,53,35)(44,62,51,56,63,59,64,61)(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);(1,3,7,18)(2,9,12,14)(4,20,22,49)(5,16,27,48)(6,28,8,17)(10,34,36,43)(11,32,24,42)(13,39,30,33)(15,29,45,23)(19,31,40,41)(21,51,52,64)(25,47,26,46)(35,59,55,62)(37,58,38,57)(44,53,63,54)(50,56,60,61);(1,4)(2,10)(3,15)(5,21)(6,29)(7,22)(8,23)(9,31)(11,35)(12,36)(13,40)(14,41)(16,44)(17,49)(18,45)(19,30)(20,28)(24,55)(25,54)(26,53)(27,52)(32,56)(33,43)(34,39)(37,50)(38,60)(42,61)(46,51)(47,64)(48,63)(57,59)(58,62);(1,5)(2,11)(3,16)(4,21)(6,26)(7,27)(8,25)(9,32)(10,35)(12,24)(13,38)(14,42)(15,44)(17,47)(18,48)(19,50)(20,51)(22,52)(23,54)(28,46)(29,53)(30,37)(31,56)(33,58)(34,59)(36,55)(39,57)(40,60)(41,61)(43,62)(45,63)(49,64)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 17411);