Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#19093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C73 |
| prime radical | #121×C73 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,7,12)(3,13,15,25)(4,17,19,26)(5,11,6,10)(8,20,23,16)(9,21,24,18)(14,22,28,32)(27,29,31,30)(33,34,38,41)(35,43,45,57)(36,40,48,55)(37,49,52,42)(39,50,54,46)(44,60,61,59)(47,58,51,56)(53,63,64,62)(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)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,25)(11,13)(12,23)(17,31)(18,30)(19,28)(21,29)(24,32)(26,27)(33,35,38,45)(34,39,41,54)(36,44,48,61)(37,50,52,46)(40,53,55,64)(42,57,49,43)(47,63,51,62)(56,59,58,60);(1,4,7,19)(2,9,12,24)(3,14,15,28)(5,18,6,21)(8,22,23,32)(10,17,11,26)(13,27,25,31)(16,29,20,30)(33,36,38,48)(34,40,41,55)(35,44,45,61)(37,51,52,47)(39,53,54,64)(42,58,49,56)(43,59,57,60)(46,63,50,62);(3,15)(8,23)(13,25)(14,28)(16,20)(22,32)(27,31)(29,30)(34,41)(35,45)(40,55)(42,49)(44,61)(46,50)(56,58)(62,63)2263 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 12081);