Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7529 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C79 · Schur multiplier (H2) = C4 |
| direct product | C79×#7531 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1408 |
| commutator | C4×C8 | C2×C4×C79 |
| Frattini, cosocle | #494 | C22×C79 |
| prime radical | #493×C79 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C79 ◁ C2×C4×C79 ◁ C42×C79 ◁ #494×C79 ◁ G | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 2 generators on 111 points:
(1,2,4,10,7,12,20,27)(3,9,14,25,16,23,30,32)(5,11,19,28,6,21,18,26)(8,17,22,15,24,29,31,13)(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);(2,8)(3,13)(5,6)(9,11)(10,25)(12,24)(14,17)(15,16)(18,19)(21,23)(22,26)(27,32)(28,31)(29,30)1975 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 519);