Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C8 |
| direct product | C79×#12514 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C79 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C79 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C42×C79 ◁ C82×C79 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18,25,32)(19,27,36,47)(20,26,39,44)(21,29,37,48)(22,28,40,45)(23,31,24,30)(33,43,38,42)(34,46,35,41)(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);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,22,33,23,34,37,39,25,36,40,38,24,35,21,20)(18,26,29,41,30,42,45,47,32,44,48,46,31,43,28,27);(2,7)(6,14)(8,12)(9,15)(17,20,37,34,24,38,22,19,25,39,21,35,23,33,40,36)(18,27,45,42,31,46,29,26,32,47,28,43,30,41,48,44)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5502);