Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C2×C79×#1419 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #357 |
| commutator | C16 | C22×C4×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| prime radical | C22×D16×C79 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C22×C8×C79 ◁ C22×C16×C79 ◁ G | |
326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(3,4,6,12,9,14,8,25)(5,15,18,32,21,10,20,26)(7,22,17,29,24,30,23,13)(11,28,27,16,31,34,33,19);(4,10)(5,16)(6,8)(7,23)(11,29)(12,32)(13,31)(14,15)(17,24)(18,28)(19,21)(20,34)(22,27)(25,26)(30,33)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6611);