Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C79 · Schur multiplier (H2) = C23 |
| direct product | C2×C79×#1400 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C22⋊C4 |
| commutator | C22 | C22×C16×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| socle | C22×C79 | C22⋊C16 |
| prime radical | C24×C79 | C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C22×C16×C79 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81);(82,83,84,89,85,90,93,101,86,91,94,102,95,103,106,111,87,88,92,98,96,99,104,108,97,100,105,109,107,110,112,113);(83,88)(84,92)(90,99)(91,100)(93,104)(94,105)(103,110)(106,112)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6543);