Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#13031 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C73 | #1879 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| prime radical | C2×#135×C73 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C73 ◁ C22×C4×C73 ◁ #405×C73 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2,6,12,31,34,29,11,8,14,28,45,7,13,5,27)(3,15,18,9,32,36,17,46,20,49,56,38,19,43,30,37)(4,21,24,42,33,44,22,41,26,39,59,10,25,40,23,47)(16,35,52,48,57,64,51,60,54,63,55,58,53,62,50,61)(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);(1,3,7,32,8,20,31,19)(2,9,34,37,14,38,13,46)(4,16,33,53,26,54,25,57)(5,18,28,17,29,56,6,30)(10,35,41,64,42,63,47,62)(11,43,45,49,27,36,12,15)(21,58,40,61,39,48,44,60)(22,55,59,51,23,52,24,50);(1,4)(2,10)(3,16)(5,24)(6,23)(7,33)(8,26)(9,35)(11,44)(12,39)(13,47)(14,42)(15,48)(17,55)(18,50)(19,57)(20,54)(21,45)(22,28)(25,31)(27,40)(29,59)(30,52)(32,53)(34,41)(36,61)(37,64)(38,63)(43,60)(46,62)(49,58)(51,56)2920 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 6019);