Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#12782 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1790 |
| commutator | C2×Q8 | C22×C4×C79 |
| Frattini, cosocle | C22×Q8 | C23×C79 |
| prime radical | C79×#3544 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C23×C79 ◁ C22×Q8×C79 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,12,7,13,5,11)(3,14,17,24,18,29,22,32)(4,10,20,9,21,27,19,28)(8,23,25,16,26,31,30,15)(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)(2,8)(4,15)(5,22)(6,17)(7,18)(9,24)(10,29)(11,25)(12,30)(13,26)(14,27)(16,21)(19,31)(20,23)(28,32);(1,4,7,21)(2,9,13,28)(3,15,18,16)(5,20,6,19)(8,24,26,32)(10,11,27,12)(14,30,29,25)(17,31,22,23)2686 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5770);