Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C2×C79×#2100 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C79 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C79 ◁ C22×C42×C79 ◁ G | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 4 generators on 105 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(3,4)(5,7)(6,8)(9,10)(11,12,14,15)(13,17,18,16);(3,5,6,9)(4,7,8,10)(12,15)(13,18)(19,20,22,25)(21,23,26,24);(4,8)(7,10)(11,13,14,18)(12,16,15,17)(20,23)(21,26)(24,25)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 16731);