Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13479 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,3,4)(5,6)(7,18)(8,14)(9,13)(10,27)(11,17)(12,21)(15,26)(16,24)(19,28)(20,30)(22,29)(23,33)(25,32)(31,36)(34,35)(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);(5,6,9,14,10,16,25,15,11,17,26,32,24,27,8,13)(7,19,21,31,22,33,35,34,23,29,36,12,28,18,20,30);(2,4)(5,7,11,23)(6,12,17,31)(8,21,25,36)(9,20,26,35)(10,28,24,22)(13,18,15,33)(14,29,32,19)(16,34,27,30)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6467);