Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33401 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | C22×D4 |
| commutator | C4 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C22×C4⋊C8 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C2×#131×C79 ◁ G | |
684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,3,10,5,11,14,7,6,12,15,20,18,9,26,21)(4,8,13,24,17,25,28,19,16,22,27,31,29,23,32,30)(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);(1,3,5,14,6,15,18,26)(2,7,9,10,12,21,11,20)(4,13,17,28,16,27,29,32)(8,19,23,24,22,30,25,31);(1,4)(2,8)(3,13)(5,17)(6,16)(7,19)(9,23)(10,24)(11,25)(12,22)(14,28)(15,27)(18,29)(20,31)(21,30)(26,32);(4,16)(8,22)(13,27)(17,29)(19,30)(23,25)(24,31)(28,32)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26389);