Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#33860 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C73 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C2×C4×C73 ◁ #443×C73 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,9,15)(3,16,21,51)(4,22,11,37)(5,26,29,25)(6,30,35,46)(7,14,8,13)(10,38,42,19)(12,33,47,34)(17,18,23,56)(20,58,36,44)(24,54,43,28)(27,61,39,40)(31,48,63,41)(32,45,64,62)(49,50,55,53)(52,57,60,59)(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);(1,3,9,21)(2,10,15,42)(4,23,11,17)(5,27,29,39)(6,31,35,63)(7,20,8,36)(12,44,47,58)(13,41,14,48)(16,46,51,30)(18,54,56,28)(19,33,38,34)(22,59,37,57)(24,61,43,40)(25,50,26,53)(32,55,64,49)(45,52,62,60);(1,4,9,11)(2,5,15,29)(3,17,21,23)(6,24,35,43)(7,26,8,25)(10,39,42,27)(12,28,47,54)(13,37,14,22)(16,49,51,55)(18,44,56,58)(19,52,38,60)(20,50,36,53)(30,32,46,64)(31,40,63,61)(33,45,34,62)(41,59,48,57);(1,5)(2,11)(3,18)(4,15)(6,32)(7,22)(8,37)(9,29)(10,40)(12,45)(13,26)(14,25)(16,50)(17,44)(19,57)(20,55)(21,56)(23,58)(24,46)(27,31)(28,34)(30,43)(33,54)(35,64)(36,49)(38,59)(39,63)(41,52)(42,61)(47,62)(48,60)(51,53)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 26848);