Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13017 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #447 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C22×D16×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
516 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,4,3)(5,6,8,14,24,33,27,15,11,17,25,13,10,16,9,26)(7,18,19,35,28,29,20,32,22,34,36,31,21,30,23,12)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3,4,2)(5,7)(6,12)(8,23)(9,19)(10,28)(11,22)(13,29)(14,30)(15,34)(16,35)(17,32)(18,26)(20,25)(21,24)(27,36)(31,33);(6,13)(8,9)(10,24)(12,29)(14,17)(15,33)(16,26)(18,35)(19,23)(20,36)(21,28)(25,27)(30,32)(31,34)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6005);