Finite Group DatabasebetaFinite Group Databasebeta| order | 18688 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C73 · Schur multiplier (H2) = C28 |
| direct product | C73×#62555 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C73 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C73 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C73 ◁ #529×C73 ◁ G | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 4672 |
A minimal faithful permutation representation — 5 generators on 121 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,23)(20,31)(21,25)(22,27)(24,33)(26,32)(28,41)(29,36)(30,37)(34,39)(35,43)(38,42)(40,45)(44,46)(47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,38)(27,37)(31,41)(32,42)(33,43)(34,44)(39,46)(40,47)(45,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,32,45,22,34,21,33)(18,24,25,39,27,40,26,31)(19,28,42,48,30,44,29,43)(23,35,36,46,37,47,38,41);(3,12)(7,10)(11,13)(15,16)(17,21,22,32)(18,25,27,26)(19,29,30,42)(20,33,34,45)(23,36,37,38)(24,39,40,31)(28,43,44,48)(35,46,47,41);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,30)(23,37)(28,44)(29,42)(35,47)(36,38)(41,46)(43,48)3796 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(18688, 55543);