Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12917 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C2×D4 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C8×D4 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C22×C16×C79 ◁ G | |
384 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)(5,6,10,16,11,15,28,33,12,17,29,34,9,25,27,18)(7,13,14,30,20,24,31,35,21,26,32,36,19,8,22,23)(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);(2,4)(5,7,10,14,11,20,28,31,12,21,29,32,9,19,27,22)(6,13,16,30,15,24,33,35,17,26,34,36,25,8,18,23);(5,8)(6,14)(7,18)(9,26)(10,23)(11,13)(12,24)(15,31)(16,20)(17,32)(19,34)(21,33)(22,25)(27,36)(28,30)(29,35)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5905);