Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#26269 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C22×D4×C79 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C4×C4⋊C4×C79 ◁ G | |
828 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)(8,9)(10,12)(13,14,19,25)(15,26,28,22)(16,20,27,21)(17,24,18,23)(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)(6,9)(10,11)(13,15,18,27,19,28,17,16)(14,20,24,22,25,21,23,26);(2,4)(13,15,18,27,19,28,17,16)(14,21,24,26,25,20,23,22);(5,7,8,12)(6,10,9,11)(13,16,18,15,19,27,17,28)(14,22,24,21,25,26,23,20)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 19257);