Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C79 · Schur multiplier (H2) = C28 |
| direct product | C79×#34085 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator | C22 | C24×C4×C79 |
| Frattini, cosocle | C23 | C25×C79 |
| socle | C23×C79 | C25 |
| prime radical | C24×C79 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
2094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 5 generators on 107 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,26)(17,20)(18,21)(19,25)(23,27)(24,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107);(1,3,4,2)(13,15)(14,19)(16,24)(17,23)(18,25)(20,28)(21,22)(26,27);(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,27,22,28);(13,16,18,17)(14,20,21,26)(15,24,25,23)(19,28,22,27);(5,7,8,11)(6,10,9,12)(13,17,18,16)(14,20,21,26)(15,23,25,24)(19,28,22,27)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 27073);