Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12918 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #120 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1850 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,7,12,5,11,21,29,8,13,24,30,6,10,23,28)(3,14,16,20,17,26,31,22,18,27,32,4,15,9,25,19)(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);(1,3,7,16,5,17,21,31,8,18,24,32,6,15,23,25)(2,9,12,19,11,14,29,20,13,26,30,22,10,27,28,4);(1,4)(2,3)(5,19)(6,22)(7,9)(8,20)(10,15)(11,17)(12,16)(13,18)(14,21)(23,27)(24,26)(25,28)(29,31)(30,32)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5906);