Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61862 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| prime radical | #449×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #533×C79 ◁ G | |
1614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 127 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26)(19,27,29,23)(20,32,34,41)(21,36,37,25)(24,33,42,35)(28,44,45,48)(30,39,31,40)(38,46,43,47)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,29)(18,23,26,27)(20,28,34,45)(21,31,37,30)(24,38,42,43)(25,40,36,39)(32,48,41,44)(33,47,35,46);(17,20)(18,24)(19,28)(21,33)(22,34)(23,38)(25,41)(26,42)(27,43)(29,45)(30,46)(31,47)(32,36)(35,37)(39,48)(40,44);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,27)(28,45)(30,31)(38,43)(39,40)(44,48)(46,47);(2,8)(6,9)(7,16)(14,15)(17,21,22,37)(18,25,26,36)(19,30,29,31)(20,33,34,35)(23,39,27,40)(24,41,42,32)(28,46,45,47)(38,48,43,44)3871 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54850);