Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #476 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1987 |
| prime radical | C23×C79 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,12,6,13,22,9,7,14,5,10,21,11,24,28)(3,15,17,26,19,29,23,27,20,25,32,31,18,30,16,8)(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)(2,8,14,27)(4,16)(5,23)(6,18)(7,20)(9,25,28,15)(10,29,12,30)(11,26,13,31)(17,24)(19,21)(22,32);(1,4,6,22,7,5,21,24)(2,9,11,12,14,28,13,10)(3,16,18,32,20,23,19,17)(8,25,26,30,27,15,31,29)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6259);