Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#12895 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C79 | C23 |
| commutator | C23 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle, prime radical | C23×C79 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C79 ◁ G | |
298 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,20,24,21,25,30,36,22,26,31,37,32,38,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(1,3,4,7)(2,5,6,8)(10,13)(14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,41)(31,39)(32,40)(36,45)(37,43)(38,44)(42,47)(46,48);(2,6)(5,8)(9,11,12,16)(10,14,13,15)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5883);