Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C2×C79×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C4×D8 |
| prime radical | C2×#508×C79 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 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);(3,4,8,13,9,15,25,14)(5,16,18,31,20,33,19,27)(6,12,21,30,22,11,7,23)(10,26,28,34,29,24,32,17);(3,5)(4,10)(6,17)(7,24)(8,18)(9,20)(11,27)(12,31)(13,28)(14,32)(15,29)(16,23)(19,25)(21,26)(22,34)(30,33);(3,6,8,21,9,22,25,7)(4,11,13,23,15,12,14,30)(5,17,18,26,20,34,19,24)(10,27,28,16,29,31,32,33)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26450);