Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C25×C4 |
| direct product | C73×#25800 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C73×#3584 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×C4×C73 ◁ G | |
1402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 4 generators on 137 points:
(1,2,7,12)(3,13,17,38)(4,9,21,33)(5,22,25,52)(6,26,23,11)(8,27,30,16)(10,34,35,24)(14,39,18,36)(15,43,45,29)(19,48,50,64)(20,51,40,32)(28,42,55,47)(31,56,57,49)(37,53,59,44)(41,54,46,62)(58,61,60,63)(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);(1,3,6,16,7,17,23,27)(2,8,11,13,12,30,26,38)(4,18,20,47,21,14,40,42)(5,15,24,44,25,45,34,53)(9,28,32,36,33,55,51,39)(10,29,22,37,35,43,52,59)(19,46,49,63,50,41,56,61)(31,54,48,58,57,62,64,60);(1,4)(2,9)(3,14,17,18)(5,19)(6,20)(7,21)(8,28,30,55)(10,31)(11,32)(12,33)(13,36,38,39)(15,41,45,46)(16,42,27,47)(22,48)(23,40)(24,49)(25,50)(26,51)(29,54,43,62)(34,56)(35,57)(37,58,59,60)(44,61,53,63)(52,64);(1,5,23,34,7,25,6,24)(2,10,26,52,12,35,11,22)(3,15,27,53,17,45,16,44)(4,19,40,56,21,50,20,49)(8,29,38,59,30,43,13,37)(9,31,51,64,33,57,32,48)(14,41,47,63,18,46,42,61)(28,54,39,60,55,62,36,58)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 18788);