Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#20560 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C24 |
| commutator | C22 | C22×C42×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C23×C4 |
| prime radical | C22×D4×C79 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 107 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 107 points:
(5,6)(7,11)(8,10)(9,12)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28)(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);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(15,21)(18,24)(22,27)(25,28);(7,12)(9,11)(13,15,16,22,17,21,23,27)(14,18,20,25,19,24,26,28);(6,10)(9,11)(14,19)(18,24)(20,26)(25,28)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 13548);