Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#9282 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #386 |
| commutator | C42 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | #532×C79 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ G | |
416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 111 points:
(2,6,8,11)(3,9,10,12)(17,18)(19,26,28,23)(20,29)(21,25)(22,24)(27,31,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,2,4,11,5,8,13,6)(3,7,12,15,10,16,9,14)(17,18,22,29,21,25,20,24)(19,23,27,31,28,26,30,32);(1,3)(2,7)(4,12)(5,10)(6,14)(8,16)(9,13)(11,15)(17,19,21,28)(18,23,25,26)(20,30,22,27)(24,32,29,31)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 2270);