Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C2×C79×#1617 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C24×C79 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C23×C4×C79 ◁ G | |
984 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(3,4,7,10)(5,8,12,15)(6,9,13,16)(11,17,18,14)(19,20,23,26)(21,27,29,34)(22,30,32,25)(24,31,33,28);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17)(19,21,23,29)(20,24,26,33)(22,31,32,28)(25,34,30,27);(3,6,7,13)(4,9,10,16)(5,11,12,18)(8,14,15,17)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11106);