Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1828 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C2×C79×#1830 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #381 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C24×C79 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
1218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 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);(82,83,86,91,87,90,100,105)(84,92,94,106,95,88,99,102)(85,96,98,110,97,89,103,104)(93,101,109,112,108,107,111,113);(82,84,87,95)(83,88,90,92)(85,93,97,108)(86,99,100,94)(89,101,96,107)(91,106,105,102)(98,111,103,109)(104,113,110,112);(82,85)(83,89,90,96)(84,93)(86,98)(87,97)(88,101,92,107)(91,104,105,110)(94,109)(95,108)(99,111)(100,103)(102,112,106,113)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 13700);