Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C73 · Schur multiplier (H2) = C24 |
| direct product | C73×#24725 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C73 |
| prime radical | C2×#41×C73 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ C2×C42×C73 ◁ G | |
658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 9344 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2,4,5)(3,7,8,6)(9,10,15,20,13,18,14,19)(11,21,30,32,23,16,24,36)(12,25,29,17,27,35,28,34)(22,38,40,37,39,33,26,31)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5)(6,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,30)(17,33)(18,21)(19,32)(20,36)(22,29)(25,37)(27,40)(28,39)(31,35)(34,38);(9,12,13,27)(10,17,18,34)(11,22,23,39)(14,29,15,28)(16,31,21,37)(19,35,20,25)(24,40,30,26)(32,38,36,33);(1,3,4,8)(2,6,5,7)(10,18)(16,21)(17,34)(19,20)(25,35)(31,37)(32,36)(33,38)3139 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 17713);