Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#22496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C22×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C22×D4×C79 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C2×C4×D4×C79 ◁ G | |
1122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 107 points:
(1,2,3,4)(6,9)(10,12)(13,14)(15,18)(16,24)(17,21)(19,25)(20,23)(22,28)(26,27)(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);(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,18,21,20)(16,25,26,22)(19,27,28,24);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,22)(17,26)(18,24)(20,27)(21,28)(23,25);(6,9)(10,12)(13,15,17,23)(14,18,21,20)(16,22,26,25)(19,24,28,27)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15484);