Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C73 · Schur multiplier (H2) = C25 |
| direct product | C73×#12746 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C73 | #418 |
| commutator | C2×C8 | C22×C4×C73 |
| Frattini, cosocle | C22×C8 | C23×C73 |
| socle | C2×C73 | #1786 |
| prime radical | C73×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C73 ◁ C22×C4×C73 ◁ C2×#131×C73 ◁ G | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 3 generators on 137 points:
(1,2,6,13,8,12,7,30)(3,14,18,43,17,44,31,39)(4,19,21,38,22,10,29,37)(5,23,26,41,28,55,27,11)(9,24,33,50,35,57,34,16)(15,42,47,64,46,60,48,63)(20,52,54,40,25,58,59,36)(32,53,61,45,51,56,62,49)(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)(2,9)(4,15)(5,24)(6,18)(7,31)(8,17)(10,32)(11,39)(12,35)(13,33)(14,23)(16,27)(19,51)(20,53)(21,47)(22,46)(25,56)(26,50)(28,57)(29,48)(30,34)(36,63)(37,61)(38,62)(40,64)(41,43)(42,52)(44,55)(45,54)(49,59)(58,60);(1,4)(2,10)(3,15)(5,25)(6,29)(7,21)(8,22)(9,32)(11,40)(12,19)(13,38)(14,42)(16,49)(17,46)(18,48)(20,28)(23,52)(24,56)(26,54)(27,59)(30,37)(31,47)(33,62)(34,61)(35,51)(36,41)(39,64)(43,63)(44,60)(45,50)(53,57)(55,58)3358 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 5734);