Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C2×C79×#1392 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #99 |
| commutator | C23 | C2×C42×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| socle | C22×C79 | #342 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
750 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);(82,83,85,90,87,89,96,102)(84,92,93,104,95,88,99,100)(86,91,97,105,98,103,109,111)(94,106,107,113,108,101,110,112);(82,84,86,94,87,95,98,108)(83,88,91,101,89,92,103,106)(85,93,97,107,96,99,109,110)(90,104,105,113,102,100,111,112)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5566);