Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13645 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C79 | C23 |
| commutator | C22 | C22×C16×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| socle | C22×C79 | C22×C16 |
| prime radical | C2×D4×C79 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C79 ◁ G | |
204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2)(3,4)(5,6,8,13,9,14,20,27,10,15,12,25,22,29,26,35,11,16,21,28,23,30,34,36,24,7,17,18,31,19,32,33)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(2,4);(5,7,8,18,9,19,20,33,10,6,12,13,22,14,26,27,11,15,21,25,23,29,34,35,24,16,17,28,31,30,32,36)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6633);