Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#61814 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C79 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ #532×C79 ◁ G | |
1688 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)(17,18,22,27)(19,23,28,36)(20,25,32,41)(21,24,34,26)(29,37,31,40)(30,35,43,38)(33,39,46,42)(44,47,45,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,28)(18,23,27,36)(20,31,32,29)(21,30,34,43)(24,35,26,38)(25,40,41,37)(33,45,46,44)(39,48,42,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(21,34)(23,35)(25,39)(26,27)(30,43)(33,46)(36,38)(37,47)(40,48)(41,42)(44,45);(3,11)(6,9)(10,16)(13,15)(19,28)(20,32)(23,36)(25,41)(30,43)(33,46)(35,38)(39,42);(2,8)(6,9)(7,14)(13,15)(17,20,22,32)(18,25,27,41)(19,29,28,31)(21,33,34,46)(23,37,36,40)(24,39,26,42)(30,44,43,45)(35,47,38,48)3871 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 54802);