Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#13281 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 123 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,14)(15,26)(16,32)(17,24)(18,23)(19,25)(20,28)(21,27)(22,33)(29,41)(30,40)(31,36)(34,35)(37,43)(38,42)(39,44)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3,4,2)(13,15,27,42,18,30,16,28,19,31,33,43,17,29,34,44)(14,20,35,40,24,38,22,26,25,39,32,41,23,37,21,36);(5,7,8,12)(6,9,10,11)(13,16,17,27,19,34,18,33)(14,21,23,32,25,22,24,35)(15,28,29,42,31,44,30,43)(20,36,37,41,39,26,38,40)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6269);