Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33369 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C2×#99 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C2×Q8 | C24×C79 |
| socle | C2×C79 | C22×#99 |
| prime radical | C79×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
1220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,10,6,11,21,28)(3,12,16,29,13,26,19,31)(4,8,17,25,18,9,20,27)(7,22,24,14,23,30,32,15)(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)(2,7)(4,14)(5,19)(6,13)(8,12)(9,26)(10,24)(11,23)(15,18)(16,21)(17,22)(20,30)(25,29)(27,31)(28,32);(3,13)(7,23)(12,26)(14,15)(16,19)(22,30)(24,32)(29,31);(1,4,6,18)(2,8,11,9)(3,14,13,15)(5,17,21,20)(7,12,23,26)(10,25,28,27)(16,30,19,22)(24,29,32,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26357);