Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33600 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2)(3,4)(5,6)(7,16)(8,12)(9,22)(10,15)(11,19)(13,17)(14,21)(18,29)(20,31)(23,27)(24,32)(25,36)(26,35)(28,33)(30,34)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(5,7,17,32,8,18,23,35,10,20,33,36,21,34,9,19)(6,11,22,30,14,25,28,31,15,26,27,29,12,24,13,16);(2,4);(5,8,10,21)(6,12,15,14)(7,18,20,34)(9,17,23,33)(11,24,26,25)(13,27,28,22)(16,29,31,30)(19,32,35,36)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26588);