Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12589 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C2×#99 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C2×#41 | C23×C79 |
| socle | C2×C79 | C2×#342 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,5,10)(3,13,16,29)(4,11,18,27)(6,9,20,26)(7,12,21,28)(8,22,24,15)(14,25,19,31)(17,23,30,32)(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);(1,3,6,17,7,14,4,15)(2,8,11,25,12,23,9,13)(5,19,20,22,21,16,18,30)(10,24,27,31,28,32,26,29);(3,14)(8,23)(13,25)(15,17)(16,19)(22,30)(24,32)(29,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5577);