Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25×C4 |
| direct product | C79×#21224 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C79 | C2×D4 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C23×C79 | C22×D4 |
| prime radical | C22×#135×C79 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C79 ◁ C22×C42×C79 ◁ G | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 103 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 103 points:
(1,2,4,3)(5,6,7,8)(9,10)(11,21)(12,24)(13,18)(14,19)(15,20)(16,22)(17,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103);(1,3,4,2)(6,8)(9,11,14,22,15,23,13,12)(10,16,18,21,20,24,19,17);(9,12,13,23,15,22,14,11)(10,17,19,24,20,21,18,16);(9,13,15,14)(10,18,20,19)(11,12,23,22)(16,21,24,17)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14212);