Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#12714 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1781 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| prime radical | C79×#3487 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×2−1+4×C79 ◁ G | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2)(3,10)(4,15,18,25)(5,19,20,9)(6,11)(7,13,22,28)(8,17,24,12)(14,26)(16,21)(23,30)(27,32)(29,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3,6,14)(2,7,11,22)(4,12,18,17)(5,13,20,28)(8,21,24,29)(9,10,19,26)(15,23,25,32)(16,27,31,30);(1,4,6,18)(2,8,11,24)(3,12,14,17)(5,16,20,31)(7,21,22,29)(9,23,19,32)(10,25,26,15)(13,27,28,30)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5702);