Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C29 |
| direct product | C2×C73×#3417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25×C73 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×#41×C73 ◁ C2×#131×C73 ◁ G | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 139 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 5 generators on 139 points:
(1,2)(67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139);(3,4,9,14)(5,15,19,43)(6,11,20,34)(7,23,27,41)(8,28,25,13)(10,30,33,17)(12,37,39,26)(16,42,44,64)(18,47,29,38)(21,50,48,63)(22,53,49,36)(24,40,55,32)(31,56,57,45)(35,60,59,52)(46,65,54,61)(51,62,66,58);(3,5,9,19)(4,10,14,33)(6,16,20,44)(7,24,27,55)(8,29,25,18)(11,31,34,57)(12,38,39,47)(13,40,28,32)(15,41,43,23)(17,37,30,26)(21,51,48,66)(22,54,49,46)(35,61,59,65)(36,62,53,58)(42,63,64,50)(45,60,56,52);(3,6)(4,11)(5,16)(7,21)(8,22)(9,20)(10,31)(12,35)(13,36)(14,34)(15,42)(17,45)(18,46)(19,44)(23,50)(24,51)(25,49)(26,52)(27,48)(28,53)(29,54)(30,56)(32,58)(33,57)(37,60)(38,61)(39,59)(40,62)(41,63)(43,64)(47,65)(55,66);(6,20)(11,34)(16,44)(21,48)(22,49)(31,57)(35,59)(36,53)(42,64)(45,56)(46,54)(50,63)(51,66)(52,60)(58,62)(61,65)4672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 55622);