Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7318 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#7318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1304 |
| commutator | C22×C4 | C42×C79 |
| Frattini, cosocle | C22×#41 | C22×C79 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C23×C79 ◁ C22×#41×C79 ◁ G | |
484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,5,9)(3,11,13,25)(4,8,16,22)(6,10,18,24)(7,19,20,12)(14,26,27,32)(15,21,28,30)(17,23,29,31)(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)(2,7,10,21)(4,15)(5,13,18,27)(6,14)(8,11,23,26)(9,20)(12,17)(16,28,29,19)(22,25)(24,30)(31,32)2212 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 306);