Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13007 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1874 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C2×#472×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,7,12,14,16,26,10,8,13,25,17,6,11,5,3)(4,18,21,15,20,30,24,32,23,29,27,31,22,9,19,28)(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);(1,3,14,17,8,10,6,12)(2,7,11,26,13,25,16,5)(4,15,20,28,23,31,22,32)(9,27,29,19,30,21,18,24);(1,4)(2,9)(3,15)(5,24)(6,22)(7,27)(8,23)(10,31)(11,29)(12,32)(13,30)(14,20)(16,18)(17,28)(19,26)(21,25)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5995);