Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#8679 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #380 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C23×C79 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 127 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18)(19,27)(20,32,34,24)(21,35)(22,26)(23,30)(25,28)(29,43,45,39)(31,40)(33,38,42,47)(36,44,46,48)(37,41)(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);(2,6)(7,15)(8,13)(9,16)(17,19,28,41,22,31,21,30)(18,23,35,40,26,37,25,27)(20,29,42,48,34,45,33,44)(24,36,47,43,32,46,38,39);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20,22,34)(18,24,26,32)(19,29,31,45)(21,33,28,42)(23,36,37,46)(25,38,35,47)(27,39,40,43)(30,44,41,48)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 1667);